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## G = S3×C24⋊C2order 288 = 25·32

### Direct product of S3 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — S3×C24⋊C2
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — S3×C24⋊C2
 Lower central C32 — C3×C6 — C3×C12 — S3×C24⋊C2
 Upper central C1 — C2 — C4 — C8

Generators and relations for S3×C24⋊C2
G = < a,b,c,d | a3=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >

Subgroups: 746 in 146 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3 [×4], C6 [×2], C6 [×4], C8, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, Dic3, Dic3 [×4], C12 [×2], C12 [×3], D6, D6 [×8], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6 [×5], C4×S3, C4×S3, D12, D12 [×4], C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3 [×2], C2×SD16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, C24⋊C2 [×5], D4.S3, Q82S3, C2×C24, C3×SD16, C2×Dic6, C2×D12, S3×D4, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C3×D12, C324Q8, C12⋊S3, C2×S32, C2×C24⋊C2, S3×SD16, D12.S3, C325SD16, S3×C24, C3×C24⋊C2, C242S3, S3×Dic6, S3×D12, S3×C24⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, D12 [×2], C22×S3 [×2], C2×SD16, S32, C24⋊C2 [×2], C2×D12, S3×D4, C2×S32, C2×C24⋊C2, S3×SD16, S3×D12, S3×C24⋊C2

Smallest permutation representation of S3×C24⋊C2
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 38)(2 25)(3 36)(4 47)(5 34)(6 45)(7 32)(8 43)(9 30)(10 41)(11 28)(12 39)(13 26)(14 37)(15 48)(16 35)(17 46)(18 33)(19 44)(20 31)(21 42)(22 29)(23 40)(24 27)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 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 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 SD16 D12 D12 C24⋊C2 S32 S3×D4 C2×S32 S3×SD16 S3×D12 S3×C24⋊C2 kernel S3×C24⋊C2 D12.S3 C32⋊5SD16 S3×C24 C3×C24⋊C2 C24⋊2S3 S3×Dic6 S3×D12 S3×C8 C24⋊C2 C3×Dic3 S3×C6 C3⋊C8 C24 Dic6 C4×S3 D12 C3×S3 Dic3 D6 S3 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 8 1 1 1 2 2 4

Matrix representation of S3×C24⋊C2 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 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 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 72 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

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

S3×C24⋊C2 in GAP, Magma, Sage, TeX

S_3\times C_{24}\rtimes C_2
% in TeX

G:=Group("S3xC24:C2");
// GroupNames label

G:=SmallGroup(288,440);
// by ID

G=gap.SmallGroup(288,440);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,58,346,80,1356,9414]);
// 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^11>;
// generators/relations

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