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## G = C8×C3⋊S3order 144 = 24·32

### Direct product of C8 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C8×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — C8×C3⋊S3
 Lower central C32 — C8×C3⋊S3
 Upper central C1 — C8

Generators and relations for C8×C3⋊S3
G = < a,b,c,d | a8=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 178 in 66 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8, C324C8, C3×C24, C4×C3⋊S3, C8×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C3⋊S3, C4×S3, C2×C3⋊S3, S3×C8, C4×C3⋊S3, C8×C3⋊S3

Smallest permutation representation of C8×C3⋊S3
On 72 points
Generators in S72
(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 15 41)(2 16 42)(3 9 43)(4 10 44)(5 11 45)(6 12 46)(7 13 47)(8 14 48)(17 56 39)(18 49 40)(19 50 33)(20 51 34)(21 52 35)(22 53 36)(23 54 37)(24 55 38)(25 66 61)(26 67 62)(27 68 63)(28 69 64)(29 70 57)(30 71 58)(31 72 59)(32 65 60)
(1 56 68)(2 49 69)(3 50 70)(4 51 71)(5 52 72)(6 53 65)(7 54 66)(8 55 67)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 27 41)(18 28 42)(19 29 43)(20 30 44)(21 31 45)(22 32 46)(23 25 47)(24 26 48)
(1 5)(2 6)(3 7)(4 8)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)

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

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

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

C8×C3⋊S3 is a maximal subgroup of
C24.60D6  C24.62D6  C48⋊S3  C3⋊S33C16  C323M5(2)  (C3×C24)⋊C4  C8⋊(C32⋊C4)  C3⋊S3.4D8  (C3×C24).C4  C8.(C32⋊C4)  S32×C8  C24⋊D6  C249D6  C244D6  C24.23D6  C24.63D6  C24.D6  D12.2D6  D245S3  C24.95D6  C24.47D6  C24.26D6  C24.40D6  C24.28D6  C12.69S32
C8×C3⋊S3 is a maximal quotient of
C48⋊S3  C12.30Dic6  C12.60D12  C12.69S32

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D6 C4×S3 S3×C8 kernel C8×C3⋊S3 C32⋊4C8 C3×C24 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C24 C12 C6 C3 # reps 1 1 1 1 2 2 8 4 4 8 16

Matrix representation of C8×C3⋊S3 in GL4(𝔽73) generated by

 51 0 0 0 0 51 0 0 0 0 46 0 0 0 0 46
,
 72 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 72 1 0 0 72 0 0 0 0 0 72 72 0 0 1 0
,
 72 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [51,0,0,0,0,51,0,0,0,0,46,0,0,0,0,46],[72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[72,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,1,1,0,0,0,0,0,1,0,0,1,0] >;

C8×C3⋊S3 in GAP, Magma, Sage, TeX

C_8\times C_3\rtimes S_3
% in TeX

G:=Group("C8xC3:S3");
// GroupNames label

G:=SmallGroup(144,85);
// by ID

G=gap.SmallGroup(144,85);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,31,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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