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## G = C3×C8⋊S3order 144 = 24·32

### Direct product of C3 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×C8⋊S3
 Lower central C3 — C6 — C3×C8⋊S3
 Upper central C1 — C12 — C24

Generators and relations for C3×C8⋊S3
G = < a,b,c,d | a3=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

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

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

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

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

C3×C8⋊S3 is a maximal subgroup of
C24⋊D6  C241D6  D24⋊S3  C24.3D6  Dic12⋊S3  C24.64D6  C24.D6  C3×S3×M4(2)  He35M4(2)  C72⋊C6  He36M4(2)
C3×C8⋊S3 is a maximal quotient of
He35M4(2)  C72⋊C6

54 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 24A ··· 24P 24Q 24R 24S 24T order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 24 ··· 24 24 24 24 24 size 1 1 6 1 1 2 2 2 1 1 6 1 1 2 2 2 6 6 2 2 6 6 1 1 1 1 2 ··· 2 6 6 2 ··· 2 6 6 6 6

54 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 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D6 M4(2) C3×S3 C4×S3 S3×C6 C8⋊S3 C3×M4(2) S3×C12 C3×C8⋊S3 kernel C3×C8⋊S3 C3×C3⋊C8 C3×C24 S3×C12 C8⋊S3 C3×Dic3 S3×C6 C3⋊C8 C24 C4×S3 Dic3 D6 C24 C12 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 4 4 4 8

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

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

C3×C8⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes S_3
% in TeX

G:=Group("C3xC8:S3");
// GroupNames label

G:=SmallGroup(144,70);
// by ID

G=gap.SmallGroup(144,70);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,313,79,69,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=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^5,d*c*d=c^-1>;
// generators/relations

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