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## G = S3×C24order 144 = 24·32

### Direct product of C24 and S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C24
G = < a,b,c | a24=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C24
On 48 points
Generators in S48
(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 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)

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

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

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

S3×C24 is a maximal subgroup of   C24.61D6  C24.63D6  C24.64D6  D6.1D12  D247S3  D6.3D12

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H 24I ··· 24T 24U ··· 24AB order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 3 3 1 1 2 2 2 1 1 3 3 1 1 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 1 1 1 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3

72 irreducible representations

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

Matrix representation of S3×C24 in GL2(𝔽73) generated by

 7 0 0 7
,
 64 0 0 8
,
 0 72 72 0
G:=sub<GL(2,GF(73))| [7,0,0,7],[64,0,0,8],[0,72,72,0] >;

S3×C24 in GAP, Magma, Sage, TeX

S_3\times C_{24}
% in TeX

G:=Group("S3xC24");
// GroupNames label

G:=SmallGroup(144,69);
// by ID

G=gap.SmallGroup(144,69);
# by ID

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

G:=Group<a,b,c|a^24=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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