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G = S3×C8.C4order 192 = 26·3

Direct product of S3 and C8.C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C8.C4, M4(2).24D6, (S3×C8).1C4, C8.31(C4×S3), C24.18(C2×C4), (C4×S3).34D4, (C2×C8).251D6, C4.210(S3×D4), C24.C46C2, D6.12(C4⋊C4), C12.369(C2×D4), C22.3(S3×Q8), (C22×S3).8Q8, Dic3.8(C4⋊C4), C12.52(C22×C4), (C2×C24).39C22, (C2×Dic3).13Q8, (S3×M4(2)).2C2, C12.53D411C2, (C2×C12).308C23, C4.Dic3.12C22, (C3×M4(2)).26C22, (S3×C2×C8).1C2, C4.82(S3×C2×C4), C6.16(C2×C4⋊C4), C2.17(S3×C4⋊C4), C31(C2×C8.C4), C3⋊C8.18(C2×C4), (C2×C6).1(C2×Q8), (C3×C8.C4)⋊2C2, (C4×S3).29(C2×C4), (C2×C3⋊C8).237C22, (S3×C2×C4).238C22, (C2×C4).411(C22×S3), SmallGroup(192,451)

Series: Derived Chief Lower central Upper central

C1C12 — S3×C8.C4
C1C3C6C12C2×C12S3×C2×C4S3×C2×C8 — S3×C8.C4
C3C6C12 — S3×C8.C4
C1C4C2×C4C8.C4

Generators and relations for S3×C8.C4
 G = < a,b,c,d | a3=b2=c8=1, d4=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 224 in 106 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], S3, C6, C6, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C8, C2×C8 [×7], M4(2) [×2], M4(2) [×4], C22×C4, C3⋊C8 [×2], C3⋊C8 [×2], C24 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C8.C4, C8.C4 [×3], C22×C8, C2×M4(2) [×2], S3×C8 [×4], S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C4.Dic3 [×2], C2×C24, C3×M4(2) [×2], S3×C2×C4, C2×C8.C4, C24.C4, C12.53D4 [×2], C3×C8.C4, S3×C2×C8, S3×M4(2) [×2], S3×C8.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×2], C22×S3, C8.C4 [×2], C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, C2×C8.C4, S3×C4⋊C4, S3×C8.C4

Smallest permutation representation of S3×C8.C4
On 48 points
Generators in S48
(1 26 45)(2 27 46)(3 28 47)(4 29 48)(5 30 41)(6 31 42)(7 32 43)(8 25 44)(9 17 39)(10 18 40)(11 19 33)(12 20 34)(13 21 35)(14 22 36)(15 23 37)(16 24 38)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)
(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 7 11 5 13 3 15)(2 16 8 10 6 12 4 14)(17 32 19 30 21 28 23 26)(18 31 20 29 22 27 24 25)(33 41 35 47 37 45 39 43)(34 48 36 46 38 44 40 42)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P12A12B12C24A24B24C24D24E24F24G24H
order12222234444446688888888888888881212122424242424242424
size1123362112336242222444466661212121222444448888

42 irreducible representations

dim111111122222222444
type++++++++--+++-
imageC1C2C2C2C2C2C4S3D4Q8Q8D6D6C4×S3C8.C4S3×D4S3×Q8S3×C8.C4
kernelS3×C8.C4C24.C4C12.53D4C3×C8.C4S3×C2×C8S3×M4(2)S3×C8C8.C4C4×S3C2×Dic3C22×S3C2×C8M4(2)C8S3C4C22C1
# reps112112812111248114

Matrix representation of S3×C8.C4 in GL4(𝔽73) generated by

1000
0100
00072
00172
,
72000
07200
0001
0010
,
22000
01000
00720
00072
,
0100
27000
0010
0001
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[22,0,0,0,0,10,0,0,0,0,72,0,0,0,0,72],[0,27,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3×C8.C4 in GAP, Magma, Sage, TeX

S_3\times C_8.C_4
% in TeX

G:=Group("S3xC8.C4");
// GroupNames label

G:=SmallGroup(192,451);
// by ID

G=gap.SmallGroup(192,451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,58,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^8=1,d^4=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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