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## G = S3×D24order 288 = 25·32

### Direct product of S3 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — S3×D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — S3×D24
 Lower central C32 — C3×C6 — C3×C12 — S3×D24
 Upper central C1 — C2 — C4 — C8

Generators and relations for S3×D24
G = < a,b,c,d | a3=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 994 in 163 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×2], S3 [×8], C6 [×2], C6 [×5], C8, C8, C2×C4, D4 [×6], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×16], C2×C6 [×3], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, D12 [×2], D12 [×8], C3⋊D4 [×2], C2×C12, C3×D4 [×2], C22×S3 [×4], C2×D8, C3×Dic3, C3×C12, S32 [×4], S3×C6, S3×C6 [×2], C2×C3⋊S3 [×2], S3×C8, D24, D24 [×5], D4⋊S3 [×2], C2×C24, C3×D8, C2×D12 [×2], S3×D4 [×2], C3×C3⋊C8, C3×C24, C3⋊D12 [×2], S3×C12, C3×D12 [×2], C12⋊S3 [×2], C2×S32 [×2], C2×D24, S3×D8, C3⋊D24 [×2], S3×C24, C3×D24, C325D8, S3×D12 [×2], S3×D24
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, D12 [×2], C22×S3 [×2], C2×D8, S32, D24 [×2], C2×D12, S3×D4, C2×S32, C2×D24, S3×D8, S3×D12, S3×D24

Smallest permutation representation of S3×D24
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)
(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 28)(2 27)(3 26)(4 25)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 3 3 12 12 36 36 2 2 4 2 6 2 2 4 6 6 24 24 2 2 6 6 2 2 4 4 4 6 6 2 2 2 2 4 ··· 4 6 6 6 6

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D8 D12 D12 D24 S32 S3×D4 C2×S32 S3×D8 S3×D12 S3×D24 kernel S3×D24 C3⋊D24 S3×C24 C3×D24 C32⋊5D8 S3×D12 S3×C8 D24 C3×Dic3 S3×C6 C3⋊C8 C24 C4×S3 D12 C3×S3 Dic3 D6 S3 C8 C6 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 1 1 1 1 2 1 2 4 2 2 8 1 1 1 2 2 4

Matrix representation of S3×D24 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 32 0 0 0 0 57 41 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 72 71 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,57,0,0,0,0,32,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72] >;

S3×D24 in GAP, Magma, Sage, TeX

S_3\times D_{24}
% in TeX

G:=Group("S3xD24");
// GroupNames label

G:=SmallGroup(288,441);
// by ID

G=gap.SmallGroup(288,441);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,142,346,80,1356,9414]);
// Polycyclic

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

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