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## G = C22×C3⋊S3order 72 = 23·32

### Direct product of C22 and C3⋊S3

Aliases: C22×C3⋊S3, C62D6, C625C2, C323C23, (C2×C6)⋊5S3, C32(C22×S3), (C3×C6)⋊3C22, SmallGroup(72,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C3⋊S3
 Chief series C1 — C3 — C32 — C3⋊S3 — C2×C3⋊S3 — C22×C3⋊S3
 Lower central C32 — C22×C3⋊S3
 Upper central C1 — C22

Generators and relations for C22×C3⋊S3
G = < a,b,c,d,e | a2=b2=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 272 in 96 conjugacy classes, 41 normal (5 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C22, C22 [×6], S3 [×16], C6 [×12], C23, C32, D6 [×24], C2×C6 [×4], C3⋊S3 [×4], C3×C6 [×3], C22×S3 [×4], C2×C3⋊S3 [×6], C62, C22×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], C22×C3⋊S3

Character table of C22×C3⋊S3

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L size 1 1 1 1 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 -1 1 -1 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 -2 2 -2 0 0 0 0 -1 2 -1 -1 1 -1 -1 1 1 -2 1 1 -1 2 1 -2 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 0 0 -1 2 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 2 -1 2 orthogonal lifted from S3 ρ11 2 -2 -2 2 0 0 0 0 -1 -1 2 -1 -2 -2 1 1 -1 -1 2 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 0 0 0 -1 -1 2 -1 2 -2 1 -1 1 1 -2 1 1 1 -1 -1 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 0 0 -1 -1 -1 2 -1 1 -2 2 1 1 1 -2 1 1 -1 -1 orthogonal lifted from D6 ρ14 2 -2 -2 2 0 0 0 0 -1 -1 -1 2 1 1 -2 -2 -1 -1 -1 2 1 1 1 1 orthogonal lifted from D6 ρ15 2 -2 2 -2 0 0 0 0 2 -1 -1 -1 1 -1 -1 1 -2 1 1 1 2 -1 -2 1 orthogonal lifted from D6 ρ16 2 -2 -2 2 0 0 0 0 2 -1 -1 -1 1 1 1 1 2 -1 -1 -1 -2 1 -2 1 orthogonal lifted from D6 ρ17 2 2 2 2 0 0 0 0 -1 -1 2 -1 2 2 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 -2 2 -2 0 0 0 0 -1 -1 2 -1 -2 2 -1 1 1 1 -2 1 -1 -1 1 1 orthogonal lifted from D6 ρ19 2 -2 2 -2 0 0 0 0 -1 -1 -1 2 1 -1 2 -2 1 1 1 -2 -1 -1 1 1 orthogonal lifted from D6 ρ20 2 2 2 2 0 0 0 0 -1 -1 -1 2 -1 -1 2 2 -1 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ21 2 2 -2 -2 0 0 0 0 -1 2 -1 -1 -1 1 1 -1 1 -2 1 1 1 -2 -1 2 orthogonal lifted from D6 ρ22 2 -2 -2 2 0 0 0 0 -1 2 -1 -1 1 1 1 1 -1 2 -1 -1 1 -2 1 -2 orthogonal lifted from D6 ρ23 2 2 2 2 0 0 0 0 2 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 2 -1 2 -1 orthogonal lifted from S3 ρ24 2 2 -2 -2 0 0 0 0 2 -1 -1 -1 -1 1 1 -1 -2 1 1 1 -2 1 2 -1 orthogonal lifted from D6

Smallest permutation representation of C22×C3⋊S3
On 36 points
Generators in S36
(1 29)(2 30)(3 28)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)
(1 11)(2 12)(3 10)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(2 3)(4 9)(5 8)(6 7)(10 12)(13 18)(14 17)(15 16)(19 21)(22 27)(23 26)(24 25)(28 30)(31 36)(32 35)(33 34)

G:=sub<Sym(36)| (1,29)(2,30)(3,28)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27), (1,11)(2,12)(3,10)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,5,8)(2,6,9)(3,4,7)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (2,3)(4,9)(5,8)(6,7)(10,12)(13,18)(14,17)(15,16)(19,21)(22,27)(23,26)(24,25)(28,30)(31,36)(32,35)(33,34)>;

G:=Group( (1,29)(2,30)(3,28)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27), (1,11)(2,12)(3,10)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,5,8)(2,6,9)(3,4,7)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (2,3)(4,9)(5,8)(6,7)(10,12)(13,18)(14,17)(15,16)(19,21)(22,27)(23,26)(24,25)(28,30)(31,36)(32,35)(33,34) );

G=PermutationGroup([(1,29),(2,30),(3,28),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27)], [(1,11),(2,12),(3,10),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,5,8),(2,6,9),(3,4,7),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(2,3),(4,9),(5,8),(6,7),(10,12),(13,18),(14,17),(15,16),(19,21),(22,27),(23,26),(24,25),(28,30),(31,36),(32,35),(33,34)])

C22×C3⋊S3 is a maximal subgroup of   C6.D12  C6.11D12  C62⋊C4  Dic3⋊D6  C22×S32  C62⋊C6
C22×C3⋊S3 is a maximal quotient of   C12.59D6  C12.D6  C12.26D6

Matrix representation of C22×C3⋊S3 in GL4(ℤ) generated by

 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1
,
 -1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 -1 1 0 0 -1 0
,
 0 -1 0 0 1 -1 0 0 0 0 -1 1 0 0 -1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,-1,0,0,1,0],[0,1,0,0,-1,-1,0,0,0,0,-1,-1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C22×C3⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes S_3
% in TeX

G:=Group("C2^2xC3:S3");
// GroupNames label

G:=SmallGroup(72,49);
// by ID

G=gap.SmallGroup(72,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,323,1204]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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