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## G = C11×3- 1+2order 297 = 33·11

### Direct product of C11 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C11×3- 1+2, C99⋊C3, C9⋊C33, C32.C33, C33.2C32, (C3×C33).C3, C3.2(C3×C33), SmallGroup(297,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C11×3- 1+2
 Chief series C1 — C3 — C33 — C99 — C11×3- 1+2
 Lower central C1 — C3 — C11×3- 1+2
 Upper central C1 — C33 — C11×3- 1+2

Generators and relations for C11×3- 1+2
G = < a,b,c | a11=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C11×3- 1+2
On 99 points
Generators in S99
(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 37 38 39 40 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55)(56 57 58 59 60 61 62 63 64 65 66)(67 68 69 70 71 72 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99)
(1 20 71 38 30 58 49 91 85)(2 21 72 39 31 59 50 92 86)(3 22 73 40 32 60 51 93 87)(4 12 74 41 33 61 52 94 88)(5 13 75 42 23 62 53 95 78)(6 14 76 43 24 63 54 96 79)(7 15 77 44 25 64 55 97 80)(8 16 67 34 26 65 45 98 81)(9 17 68 35 27 66 46 99 82)(10 18 69 36 28 56 47 89 83)(11 19 70 37 29 57 48 90 84)
(12 94 33)(13 95 23)(14 96 24)(15 97 25)(16 98 26)(17 99 27)(18 89 28)(19 90 29)(20 91 30)(21 92 31)(22 93 32)(56 83 69)(57 84 70)(58 85 71)(59 86 72)(60 87 73)(61 88 74)(62 78 75)(63 79 76)(64 80 77)(65 81 67)(66 82 68)

G:=sub<Sym(99)| (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)(34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99), (1,20,71,38,30,58,49,91,85)(2,21,72,39,31,59,50,92,86)(3,22,73,40,32,60,51,93,87)(4,12,74,41,33,61,52,94,88)(5,13,75,42,23,62,53,95,78)(6,14,76,43,24,63,54,96,79)(7,15,77,44,25,64,55,97,80)(8,16,67,34,26,65,45,98,81)(9,17,68,35,27,66,46,99,82)(10,18,69,36,28,56,47,89,83)(11,19,70,37,29,57,48,90,84), (12,94,33)(13,95,23)(14,96,24)(15,97,25)(16,98,26)(17,99,27)(18,89,28)(19,90,29)(20,91,30)(21,92,31)(22,93,32)(56,83,69)(57,84,70)(58,85,71)(59,86,72)(60,87,73)(61,88,74)(62,78,75)(63,79,76)(64,80,77)(65,81,67)(66,82,68)>;

G:=Group( (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)(34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99), (1,20,71,38,30,58,49,91,85)(2,21,72,39,31,59,50,92,86)(3,22,73,40,32,60,51,93,87)(4,12,74,41,33,61,52,94,88)(5,13,75,42,23,62,53,95,78)(6,14,76,43,24,63,54,96,79)(7,15,77,44,25,64,55,97,80)(8,16,67,34,26,65,45,98,81)(9,17,68,35,27,66,46,99,82)(10,18,69,36,28,56,47,89,83)(11,19,70,37,29,57,48,90,84), (12,94,33)(13,95,23)(14,96,24)(15,97,25)(16,98,26)(17,99,27)(18,89,28)(19,90,29)(20,91,30)(21,92,31)(22,93,32)(56,83,69)(57,84,70)(58,85,71)(59,86,72)(60,87,73)(61,88,74)(62,78,75)(63,79,76)(64,80,77)(65,81,67)(66,82,68) );

G=PermutationGroup([[(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),(34,35,36,37,38,39,40,41,42,43,44),(45,46,47,48,49,50,51,52,53,54,55),(56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99)], [(1,20,71,38,30,58,49,91,85),(2,21,72,39,31,59,50,92,86),(3,22,73,40,32,60,51,93,87),(4,12,74,41,33,61,52,94,88),(5,13,75,42,23,62,53,95,78),(6,14,76,43,24,63,54,96,79),(7,15,77,44,25,64,55,97,80),(8,16,67,34,26,65,45,98,81),(9,17,68,35,27,66,46,99,82),(10,18,69,36,28,56,47,89,83),(11,19,70,37,29,57,48,90,84)], [(12,94,33),(13,95,23),(14,96,24),(15,97,25),(16,98,26),(17,99,27),(18,89,28),(19,90,29),(20,91,30),(21,92,31),(22,93,32),(56,83,69),(57,84,70),(58,85,71),(59,86,72),(60,87,73),(61,88,74),(62,78,75),(63,79,76),(64,80,77),(65,81,67),(66,82,68)]])

121 conjugacy classes

 class 1 3A 3B 3C 3D 9A ··· 9F 11A ··· 11J 33A ··· 33T 33U ··· 33AN 99A ··· 99BH order 1 3 3 3 3 9 ··· 9 11 ··· 11 33 ··· 33 33 ··· 33 99 ··· 99 size 1 1 1 3 3 3 ··· 3 1 ··· 1 1 ··· 1 3 ··· 3 3 ··· 3

121 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C11 C33 C33 3- 1+2 C11×3- 1+2 kernel C11×3- 1+2 C99 C3×C33 3- 1+2 C9 C32 C11 C1 # reps 1 6 2 10 60 20 2 20

Matrix representation of C11×3- 1+2 in GL3(𝔽199) generated by

 61 0 0 0 61 0 0 0 61
,
 106 91 0 92 93 92 0 107 0
,
 1 0 0 106 92 0 198 0 106
G:=sub<GL(3,GF(199))| [61,0,0,0,61,0,0,0,61],[106,92,0,91,93,107,0,92,0],[1,106,198,0,92,0,0,0,106] >;

C11×3- 1+2 in GAP, Magma, Sage, TeX

C_{11}\times 3_-^{1+2}
% in TeX

G:=Group("C11xES-(3,1)");
// GroupNames label

G:=SmallGroup(297,4);
// by ID

G=gap.SmallGroup(297,4);
# by ID

G:=PCGroup([4,-3,-3,-11,-3,396,817]);
// Polycyclic

G:=Group<a,b,c|a^11=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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