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## G = C2×C6.S32order 432 = 24·33

### Direct product of C2 and C6.S32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×C6.S32
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C6.S32 — C2×C6.S32
 Lower central He3 — C2×C6.S32
 Upper central C1 — C22

Generators and relations for C2×C6.S32
G = < a,b,c,d,e | a2=b3=c3=d6=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, dbd-1=b-1c-1, dcd-1=ece-1=c-1, ede-1=d-1 >

Subgroups: 1019 in 205 conjugacy classes, 61 normal (21 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, S3×C2×C4, C22×Dic3, C32⋊C6, C2×He3, C2×He3, S3×Dic3, C6.D6, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C32⋊C12, He33C4, C2×C32⋊C6, C22×He3, C2×S3×Dic3, C2×C6.D6, C6.S32, C2×C32⋊C12, C2×He33C4, C22×C32⋊C6, C2×C6.S32
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, S32, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, C32⋊D6, C2×S3×Dic3, C6.S32, C2×C32⋊D6, C2×C6.S32

Smallest permutation representation of C2×C6.S32
On 72 points
Generators in S72
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 66)(8 61)(9 62)(10 63)(11 64)(12 65)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(1 26 22)(2 27 23)(4 19 29)(5 20 30)(8 71 14)(9 72 15)(11 17 68)(12 18 69)(31 41 48)(33 44 37)(34 45 38)(36 40 47)(50 64 57)(51 65 58)(53 60 61)(54 55 62)
(1 26 22)(2 23 27)(3 28 24)(4 19 29)(5 30 20)(6 21 25)(7 13 70)(8 71 14)(9 15 72)(10 67 16)(11 17 68)(12 69 18)(31 48 41)(32 42 43)(33 44 37)(34 38 45)(35 46 39)(36 40 47)(49 56 63)(50 64 57)(51 58 65)(52 66 59)(53 60 61)(54 62 55)
(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)
(1 8 4 11)(2 7 5 10)(3 12 6 9)(13 20 67 27)(14 19 68 26)(15 24 69 25)(16 23 70 30)(17 22 71 29)(18 21 72 28)(31 63 34 66)(32 62 35 65)(33 61 36 64)(37 53 47 57)(38 52 48 56)(39 51 43 55)(40 50 44 60)(41 49 45 59)(42 54 46 58)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A ··· 4H 6A 6B 6C 6D ··· 6I 6J 6K 6L 6M 6N 6O 6P 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 3 4 ··· 4 6 6 6 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 size 1 1 1 1 9 9 9 9 2 6 6 12 9 ··· 9 2 2 2 6 ··· 6 12 12 12 18 18 18 18 18 ··· 18

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 6 6 6 type + + + + + + + + - + + + - + + + image C1 C2 C2 C2 C2 C4 S3 S3 D6 Dic3 D6 D6 C4×S3 S32 S3×Dic3 C2×S32 C32⋊D6 C6.S32 C2×C32⋊D6 kernel C2×C6.S32 C6.S32 C2×C32⋊C12 C2×He3⋊3C4 C22×C32⋊C6 C2×C32⋊C6 C2×C3⋊Dic3 C22×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C2×C3⋊S3 C62 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 1 2 4 2 2 4 1 2 1 2 4 2

Matrix representation of C2×C6.S32 in GL10(𝔽13)

 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12
,
 0 0 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 5 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 5 5 0 0

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,12,1,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,12,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0] >;

C2×C6.S32 in GAP, Magma, Sage, TeX

C_2\times C_6.S_3^2
% in TeX

G:=Group("C2xC6.S3^2");
// GroupNames label

G:=SmallGroup(432,317);
// by ID

G=gap.SmallGroup(432,317);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,571,4037,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^6=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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