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G = C2xC8:S3order 96 = 25·3

Direct product of C2 and C8:S3

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

Aliases: C2xC8:S3, C8:9D6, C6:1M4(2), C24:12C22, C12.36C23, (C2xC8):6S3, C4o(C8:S3), (C2xC24):11C2, (C4xS3).3C4, C4.24(C4xS3), C3:C8:10C22, D6.5(C2xC4), (C2xC4).98D6, C3:1(C2xM4(2)), C12.27(C2xC4), (C22xS3).3C4, C4.36(C22xS3), C6.13(C22xC4), C22.14(C4xS3), (C2xDic3).5C4, Dic3.6(C2xC4), (C4xS3).14C22, (C2xC12).111C22, (C2xC3:C8):11C2, C2.14(S3xC2xC4), (S3xC2xC4).10C2, (C2xC6).15(C2xC4), SmallGroup(96,107)

Series: Derived Chief Lower central Upper central

C1C6 — C2xC8:S3
C1C3C6C12C4xS3S3xC2xC4 — C2xC8:S3
C3C6 — C2xC8:S3
C1C2xC4C2xC8

Generators and relations for C2xC8:S3
 G = < a,b,c,d | a2=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 130 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, C23, Dic3, C12, D6, D6, C2xC6, C2xC8, C2xC8, M4(2), C22xC4, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C22xS3, C2xM4(2), C8:S3, C2xC3:C8, C2xC24, S3xC2xC4, C2xC8:S3
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, M4(2), C22xC4, C4xS3, C22xS3, C2xM4(2), C8:S3, S3xC2xC4, C2xC8:S3

Smallest permutation representation of C2xC8:S3
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(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 45 9)(2 46 10)(3 47 11)(4 48 12)(5 41 13)(6 42 14)(7 43 15)(8 44 16)(17 26 37)(18 27 38)(19 28 39)(20 29 40)(21 30 33)(22 31 34)(23 32 35)(24 25 36)
(2 6)(4 8)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(17 21)(19 23)(25 36)(26 33)(27 38)(28 35)(29 40)(30 37)(31 34)(32 39)

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

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

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

C2xC8:S3 is a maximal subgroup of
C8.25D12  C8:6D12  D6.C42  C8:9D12  Dic3:5M4(2)  D6.4C42  D6:M4(2)  D6:C8:C2  Dic3:M4(2)  C3:C8:26D4  C4:C4:19D6  D4:(C4xS3)  C3:C8:1D4  C3:C8:D4  (S3xQ8):C4  Q8:7(C4xS3)  C3:(C8:D4)  C3:C8.D4  D6:3M4(2)  C12:M4(2)  C12:2M4(2)  C42.30D6  C8:(C4xS3)  C24:7D4  C8.2D12  C8:S3:C4  C8:3D12  M4(2).25D6  C24:33D4  C24:21D4  C24:12D4  C24:8D4  C24.36D4  C2xS3xM4(2)  M4(2):28D6  SD16:D6  C5:C8:D6
C2xC8:S3 is a maximal quotient of
C24:12Q8  C42.282D6  C8:6D12  Dic3.M4(2)  D6:M4(2)  C3:C8:26D4  C42.198D6  C42.202D6  C12:M4(2)  C12:2M4(2)  C24:33D4  C5:C8:D6

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D24A···24H
order1222223444444666888888881212121224···24
size11116621111662222222666622222···2

36 irreducible representations

dim111111112222222
type++++++++
imageC1C2C2C2C2C4C4C4S3D6D6M4(2)C4xS3C4xS3C8:S3
kernelC2xC8:S3C8:S3C2xC3:C8C2xC24S3xC2xC4C4xS3C2xDic3C22xS3C2xC8C8C2xC4C6C4C22C2
# reps141114221214228

Matrix representation of C2xC8:S3 in GL3(F73) generated by

7200
010
001
,
7200
06516
0578
,
100
0721
0720
,
7200
001
010
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[72,0,0,0,65,57,0,16,8],[1,0,0,0,72,72,0,1,0],[72,0,0,0,0,1,0,1,0] >;

C2xC8:S3 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes S_3
% in TeX

G:=Group("C2xC8:S3");
// GroupNames label

G:=SmallGroup(96,107);
// by ID

G=gap.SmallGroup(96,107);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,50,69,2309]);
// Polycyclic

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

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