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## G = C2×C12.29D6order 288 = 25·32

### Direct product of C2 and C12.29D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C12.29D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.29D6 — C2×C12.29D6
 Lower central C32 — C2×C12.29D6
 Upper central C1 — C2×C4

Generators and relations for C2×C12.29D6
G = < a,b,c,d | a2=b12=1, c6=b3, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c5 >

Subgroups: 594 in 179 conjugacy classes, 68 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×18], C2×C6 [×2], C2×C6, C2×C8 [×6], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×2], C2×C12, C22×S3 [×3], C22×C8, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C8 [×8], C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×3], C3×C3⋊C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C2×C8 [×2], C12.29D6 [×4], C6×C3⋊C8 [×2], C2×C4×C3⋊S3, C2×C12.29D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C8 [×4], C2×C4 [×6], C23, D6 [×6], C2×C8 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C22×C8, S32, S3×C8 [×4], S3×C2×C4 [×2], C6.D6 [×2], C2×S32, S3×C2×C8 [×2], C12.29D6 [×2], C2×C6.D6, C2×C12.29D6

Smallest permutation representation of C2×C12.29D6
On 48 points
Generators in S48
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
(1 3 5 7 9 11 13 15 17 19 21 23)(2 12 22 8 18 4 14 24 10 20 6 16)(25 27 29 31 33 35 37 39 41 43 45 47)(26 36 46 32 42 28 38 48 34 44 30 40)
(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)
(1 15 13 3)(2 8 14 20)(4 18 16 6)(5 11 17 23)(7 21 19 9)(10 24 22 12)(25 39 37 27)(26 32 38 44)(28 42 40 30)(29 35 41 47)(31 45 43 33)(34 48 46 36)

G:=sub<Sym(48)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36), (1,3,5,7,9,11,13,15,17,19,21,23)(2,12,22,8,18,4,14,24,10,20,6,16)(25,27,29,31,33,35,37,39,41,43,45,47)(26,36,46,32,42,28,38,48,34,44,30,40), (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), (1,15,13,3)(2,8,14,20)(4,18,16,6)(5,11,17,23)(7,21,19,9)(10,24,22,12)(25,39,37,27)(26,32,38,44)(28,42,40,30)(29,35,41,47)(31,45,43,33)(34,48,46,36)>;

G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36), (1,3,5,7,9,11,13,15,17,19,21,23)(2,12,22,8,18,4,14,24,10,20,6,16)(25,27,29,31,33,35,37,39,41,43,45,47)(26,36,46,32,42,28,38,48,34,44,30,40), (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), (1,15,13,3)(2,8,14,20)(4,18,16,6)(5,11,17,23)(7,21,19,9)(10,24,22,12)(25,39,37,27)(26,32,38,44)(28,42,40,30)(29,35,41,47)(31,45,43,33)(34,48,46,36) );

G=PermutationGroup([(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36)], [(1,3,5,7,9,11,13,15,17,19,21,23),(2,12,22,8,18,4,14,24,10,20,6,16),(25,27,29,31,33,35,37,39,41,43,45,47),(26,36,46,32,42,28,38,48,34,44,30,40)], [(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)], [(1,15,13,3),(2,8,14,20),(4,18,16,6),(5,11,17,23),(7,21,19,9),(10,24,22,12),(25,39,37,27),(26,32,38,44),(28,42,40,30),(29,35,41,47),(31,45,43,33),(34,48,46,36)])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 8A ··· 8P 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 9 9 9 9 2 2 4 1 1 1 1 9 9 9 9 2 ··· 2 4 4 4 3 ··· 3 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 S3 D6 D6 C4×S3 C4×S3 S3×C8 S32 C6.D6 C2×S32 C6.D6 C12.29D6 kernel C2×C12.29D6 C12.29D6 C6×C3⋊C8 C2×C4×C3⋊S3 C4×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C2×C3⋊S3 C2×C3⋊C8 C3⋊C8 C2×C12 C12 C2×C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 16 2 4 2 4 4 16 1 1 1 1 4

Matrix representation of C2×C12.29D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 22 0 0 0 0 51 22 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 27 46 0 0 0 0 0 46 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,51,0,0,0,0,22,22,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[27,0,0,0,0,0,46,46,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C12.29D6 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{29}D_6
% in TeX

G:=Group("C2xC12.29D6");
// GroupNames label

G:=SmallGroup(288,464);
// by ID

G=gap.SmallGroup(288,464);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^6=b^3,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^5>;
// generators/relations

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