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## G = C6×C3⋊C8order 144 = 24·32

### Direct product of C6 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8 — C6×C3⋊C8
 Lower central C3 — C6×C3⋊C8
 Upper central C1 — C2×C12

Generators and relations for C6×C3⋊C8
G = < a,b,c | a6=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C6×C3⋊C8
On 48 points
Generators in S48
(1 33 26 15 18 47)(2 34 27 16 19 48)(3 35 28 9 20 41)(4 36 29 10 21 42)(5 37 30 11 22 43)(6 38 31 12 23 44)(7 39 32 13 24 45)(8 40 25 14 17 46)
(1 26 18)(2 19 27)(3 28 20)(4 21 29)(5 30 22)(6 23 31)(7 32 24)(8 17 25)(9 41 35)(10 36 42)(11 43 37)(12 38 44)(13 45 39)(14 40 46)(15 47 33)(16 34 48)
(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)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 C3×C3⋊C8 kernel C6×C3⋊C8 C3×C3⋊C8 C6×C12 C2×C3⋊C8 C3×C12 C62 C3⋊C8 C2×C12 C3×C6 C12 C2×C6 C6 C2×C12 C12 C12 C2×C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 1 1 1 2 4 2 2 2 8

Matrix representation of C6×C3⋊C8 in GL4(𝔽73) generated by

 65 0 0 0 0 65 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 8 7 0 0 0 64
,
 63 0 0 0 0 72 0 0 0 0 63 4 0 0 66 10
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,7,64],[63,0,0,0,0,72,0,0,0,0,63,66,0,0,4,10] >;

C6×C3⋊C8 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes C_8
% in TeX

G:=Group("C6xC3:C8");
// GroupNames label

G:=SmallGroup(144,74);
// by ID

G=gap.SmallGroup(144,74);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,69,3461]);
// Polycyclic

G:=Group<a,b,c|a^6=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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