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## G = S3×C8⋊S3order 288 = 25·32

### Direct product of S3 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×C8⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — C4×S32 — S3×C8⋊S3
 Lower central C32 — C3×C6 — S3×C8⋊S3
 Upper central C1 — C4 — C8

Generators and relations for S3×C8⋊S3
G = < a,b,c,d,e | a3=b2=c8=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >

Subgroups: 498 in 146 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3 [×4], C6 [×2], C6 [×4], C8, C8 [×3], C2×C4 [×6], C23, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×7], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3×S3 [×2], C3×S3, C3⋊S3, C3×C6, C3⋊C8 [×2], C3⋊C8 [×3], C24 [×2], C24 [×3], C4×S3 [×2], C4×S3 [×7], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×M4(2), C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8, S3×C8, C8⋊S3, C8⋊S3 [×6], C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4 [×2], C3×C3⋊C8 [×2], C324C8, C3×C24, S3×Dic3 [×2], C6.D6, S3×C12 [×2], C4×C3⋊S3, C2×S32, C2×C8⋊S3, S3×M4(2), S3×C3⋊C8, D6.Dic3, C12.31D6, S3×C24, C3×C8⋊S3, C24⋊S3, C4×S32, S3×C8⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], M4(2) [×2], C22×C4, C4×S3 [×4], C22×S3 [×2], C2×M4(2), S32, C8⋊S3 [×2], S3×C2×C4 [×2], C2×S32, C2×C8⋊S3, S3×M4(2), C4×S32, S3×C8⋊S3

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 12G 12H 12I 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N 24O 24P order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 24 24 size 1 1 3 3 6 18 2 2 4 1 1 3 3 6 18 2 2 4 6 6 12 2 2 6 6 6 6 18 18 2 2 2 2 4 4 6 6 12 2 2 2 2 4 ··· 4 6 6 6 6 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 S3 S3 D6 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 S32 C2×S32 S3×M4(2) C4×S32 S3×C8⋊S3 kernel S3×C8⋊S3 S3×C3⋊C8 D6.Dic3 C12.31D6 S3×C24 C3×C8⋊S3 C24⋊S3 C4×S32 S3×Dic3 C6.D6 C2×S32 S3×C8 C8⋊S3 C3⋊C8 C24 C4×S3 C3×S3 Dic3 D6 S3 C8 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 4 2 2 1 1 2 2 2 4 4 4 8 1 1 2 2 4

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

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

S3×C8⋊S3 in GAP, Magma, Sage, TeX

S_3\times C_8\rtimes S_3
% in TeX

G:=Group("S3xC8:S3");
// GroupNames label

G:=SmallGroup(288,438);
// by ID

G=gap.SmallGroup(288,438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations

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