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G = S3×C8⋊S3order 288 = 25·32

Direct product of S3 and C8⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: S3×C8⋊S3, C2420D6, C86S32, C3⋊C826D6, (S3×C8)⋊7S3, D6.4(C4×S3), (S3×C24)⋊13C2, C24⋊S37C2, (C4×S3).27D6, C31(S3×M4(2)), (C3×C24)⋊20C22, (C3×S3)⋊1M4(2), D6.Dic39C2, Dic3.7(C4×S3), (S3×Dic3).2C4, C6.D6.2C4, C321(C2×M4(2)), C12.31D69C2, (S3×C12).1C22, C12.135(C22×S3), (C3×C12).136C23, C324C814C22, C2.4(C4×S32), C6.2(S3×C2×C4), (S3×C3⋊C8)⋊10C2, (C4×S32).1C2, (C2×S32).3C4, C4.82(C2×S32), C31(C2×C8⋊S3), (C3×C8⋊S3)⋊9C2, (C3×C3⋊C8)⋊22C22, (S3×C6).9(C2×C4), (C3×C6).2(C22×C4), (C4×C3⋊S3).57C22, C3⋊Dic3.17(C2×C4), (C3×Dic3).1(C2×C4), (C2×C3⋊S3).13(C2×C4), SmallGroup(288,438)

Series: Derived Chief Lower central Upper central

C1C3×C6 — S3×C8⋊S3
C1C3C32C3×C6C3×C12S3×C12C4×S32 — S3×C8⋊S3
C32C3×C6 — S3×C8⋊S3
C1C4C8

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 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A6B6C6D6E6F8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H12I24A24B24C24D24E···24J24K24L24M24N24O24P
order122222333444444666666888888881212121212121212122424242424···24242424242424
size1133618224113361822466122266661818222244661222224···466661212

54 irreducible representations

dim1111111111122222222244444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4S3S3D6D6D6M4(2)C4×S3C4×S3C8⋊S3S32C2×S32S3×M4(2)C4×S32S3×C8⋊S3
kernelS3×C8⋊S3S3×C3⋊C8D6.Dic3C12.31D6S3×C24C3×C8⋊S3C24⋊S3C4×S32S3×Dic3C6.D6C2×S32S3×C8C8⋊S3C3⋊C8C24C4×S3C3×S3Dic3D6S3C8C4C3C2C1
# reps1111111142211222444811224

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

4403
2040
0441
2030
,
0403
3010
0401
3020
,
1020
0102
1040
0104
,
3020
0302
1010
0101
,
4020
0103
0010
0004
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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