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## G = S3×C3⋊C8order 144 = 24·32

### Direct product of S3 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C3⋊C8
G = < a,b,c,d | a3=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of S3×C3⋊C8
On 48 points
Generators in S48
(1 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 23 44)(10 24 45)(11 17 46)(12 18 47)(13 19 48)(14 20 41)(15 21 42)(16 22 43)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(1 35 25)(2 26 36)(3 37 27)(4 28 38)(5 39 29)(6 30 40)(7 33 31)(8 32 34)(9 23 44)(10 45 24)(11 17 46)(12 47 18)(13 19 48)(14 41 20)(15 21 42)(16 43 22)
(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)| (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,23,44)(10,24,45)(11,17,46)(12,18,47)(13,19,48)(14,20,41)(15,21,42)(16,22,43), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (1,35,25)(2,26,36)(3,37,27)(4,28,38)(5,39,29)(6,30,40)(7,33,31)(8,32,34)(9,23,44)(10,45,24)(11,17,46)(12,47,18)(13,19,48)(14,41,20)(15,21,42)(16,43,22), (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( (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,23,44)(10,24,45)(11,17,46)(12,18,47)(13,19,48)(14,20,41)(15,21,42)(16,22,43), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (1,35,25)(2,26,36)(3,37,27)(4,28,38)(5,39,29)(6,30,40)(7,33,31)(8,32,34)(9,23,44)(10,45,24)(11,17,46)(12,47,18)(13,19,48)(14,41,20)(15,21,42)(16,43,22), (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([[(1,25,35),(2,26,36),(3,27,37),(4,28,38),(5,29,39),(6,30,40),(7,31,33),(8,32,34),(9,23,44),(10,24,45),(11,17,46),(12,18,47),(13,19,48),(14,20,41),(15,21,42),(16,22,43)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(1,35,25),(2,26,36),(3,37,27),(4,28,38),(5,39,29),(6,30,40),(7,33,31),(8,32,34),(9,23,44),(10,45,24),(11,17,46),(12,47,18),(13,19,48),(14,41,20),(15,21,42),(16,43,22)], [(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)]])

S3×C3⋊C8 is a maximal subgroup of
S32×C8  C24.64D6  C24.D6  D12.2Dic3  D12.Dic3  D12.22D6  Dic6.20D6  D12.12D6  D12.13D6  C32⋊C6⋊C8  C12.89S32  C12.93S32
S3×C3⋊C8 is a maximal quotient of
C24.61D6  C12.77D12  C12.81D12  C32⋊C6⋊C8  C12.93S32

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C8 S3 S3 Dic3 D6 Dic3 C3⋊C8 C4×S3 S3×C8 S32 S3×Dic3 S3×C3⋊C8 kernel S3×C3⋊C8 C3×C3⋊C8 C32⋊4C8 S3×C12 C3×Dic3 S3×C6 C3×S3 C3⋊C8 C4×S3 Dic3 C12 D6 S3 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 2 8 1 1 1 2 1 4 2 4 1 1 2

Matrix representation of S3×C3⋊C8 in GL4(𝔽5) generated by

 1 2 0 0 1 3 0 0 0 0 3 3 0 0 4 1
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 1 2 0 0 1 3 0 0 0 0 1 2 0 0 1 3
,
 0 0 0 3 0 0 1 0 0 3 0 0 1 0 0 0
G:=sub<GL(4,GF(5))| [1,1,0,0,2,3,0,0,0,0,3,4,0,0,3,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,1,0,0,2,3,0,0,0,0,1,1,0,0,2,3],[0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,0] >;

S3×C3⋊C8 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes C_8
% in TeX

G:=Group("S3xC3:C8");
// GroupNames label

G:=SmallGroup(144,52);
// by ID

G=gap.SmallGroup(144,52);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,31,50,490,3461]);
// Polycyclic

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

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