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## G = C2×C9⋊C12order 216 = 23·33

### Direct product of C2 and C9⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C9⋊C12
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C9⋊C12 — C2×C9⋊C12
 Lower central C9 — C2×C9⋊C12
 Upper central C1 — C22

Generators and relations for C2×C9⋊C12
G = < a,b,c | a2=b9=c12=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C2×C9⋊C12
On 72 points
Generators in S72
(1 12)(2 9)(3 10)(4 11)(5 23)(6 24)(7 21)(8 22)(13 17)(14 18)(15 19)(16 20)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 61)(59 62)(60 63)
(1 39 63 18 47 67 6 43 71)(2 68 40 7 64 44 19 72 48)(3 45 69 20 41 61 8 37 65)(4 62 46 5 70 38 17 66 42)(9 53 29 21 49 33 15 57 25)(10 34 54 16 30 58 22 26 50)(11 59 35 23 55 27 13 51 31)(12 28 60 14 36 52 24 32 56)
(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)

G:=sub<Sym(72)| (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,17)(14,18)(15,19)(16,20)(25,48)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,61)(59,62)(60,63), (1,39,63,18,47,67,6,43,71)(2,68,40,7,64,44,19,72,48)(3,45,69,20,41,61,8,37,65)(4,62,46,5,70,38,17,66,42)(9,53,29,21,49,33,15,57,25)(10,34,54,16,30,58,22,26,50)(11,59,35,23,55,27,13,51,31)(12,28,60,14,36,52,24,32,56), (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)>;

G:=Group( (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,17)(14,18)(15,19)(16,20)(25,48)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,61)(59,62)(60,63), (1,39,63,18,47,67,6,43,71)(2,68,40,7,64,44,19,72,48)(3,45,69,20,41,61,8,37,65)(4,62,46,5,70,38,17,66,42)(9,53,29,21,49,33,15,57,25)(10,34,54,16,30,58,22,26,50)(11,59,35,23,55,27,13,51,31)(12,28,60,14,36,52,24,32,56), (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) );

G=PermutationGroup([[(1,12),(2,9),(3,10),(4,11),(5,23),(6,24),(7,21),(8,22),(13,17),(14,18),(15,19),(16,20),(25,48),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,61),(59,62),(60,63)], [(1,39,63,18,47,67,6,43,71),(2,68,40,7,64,44,19,72,48),(3,45,69,20,41,61,8,37,65),(4,62,46,5,70,38,17,66,42),(9,53,29,21,49,33,15,57,25),(10,34,54,16,30,58,22,26,50),(11,59,35,23,55,27,13,51,31),(12,28,60,14,36,52,24,32,56)], [(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)]])

C2×C9⋊C12 is a maximal subgroup of   Dic9⋊C12  C36⋊C12  D18⋊C12  C62.27D6  C2×C4×C9⋊C6  Dic182C6
C2×C9⋊C12 is a maximal quotient of   C36.C12  C36⋊C12  C62.27D6

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D ··· 6I 9A 9B 9C 12A ··· 12H 18A ··· 18I order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 size 1 1 1 1 2 3 3 9 9 9 9 2 2 2 3 ··· 3 6 6 6 9 ··· 9 6 ··· 6

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 type + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 C9⋊C6 C9⋊C12 C2×C9⋊C6 kernel C2×C9⋊C12 C9⋊C12 C22×3- 1+2 C2×Dic9 C2×3- 1+2 Dic9 C2×C18 C18 C62 C3×C6 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 2 4 2 1 2 1

Matrix representation of C2×C9⋊C12 in GL8(𝔽37)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36
,
 36 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26 0
,
 29 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 10 0 0 36 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 27 0 0 0

G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36],[36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0],[29,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,10,0,0,0] >;

C2×C9⋊C12 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes C_{12}
% in TeX

G:=Group("C2xC9:C12");
// GroupNames label

G:=SmallGroup(216,61);
// by ID

G=gap.SmallGroup(216,61);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,3604,736,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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