Copied to
clipboard

G = S3xC2xC8order 96 = 25·3

Direct product of C2xC8 and S3

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

Aliases: S3xC2xC8, C24:11C22, C12.35C23, C8o(S3xC8), C4o(S3xC8), C8o(C4xS3), C6:1(C2xC8), C3:1(C22xC8), Dic3o(C2xC8), C8o(C2xDic3), (C2xC24):10C2, (C4xS3).5C4, C4.23(C4xS3), C3:C8:13C22, D6.9(C2xC4), (C2xC4).97D6, C12.26(C2xC4), (C22xS3).5C4, C4.35(C22xS3), C6.12(C22xC4), C22.13(C4xS3), (C2xDic3).8C4, (C4xS3).17C22, Dic3.10(C2xC4), (C2xC12).110C22, C8o2(C2xC3:C8), C2.2(S3xC2xC4), (C2xC3:C8):13C2, (C2xC8)o2(C3:C8), (S3xC2xC4).12C2, (C2xC8)o(C2xDic3), (C2xC6).14(C2xC4), (C2xC8)o(C2xC3:C8), SmallGroup(96,106)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC2xC8
C1C3C6C12C4xS3S3xC2xC4 — S3xC2xC8
C3 — S3xC2xC8
C1C2xC8

Generators and relations for S3xC2xC8
 G = < a,b,c,d | a2=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 130 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, C23, Dic3, C12, D6, C2xC6, C2xC8, C2xC8, C22xC4, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C22xS3, C22xC8, S3xC8, C2xC3:C8, C2xC24, S3xC2xC4, S3xC2xC8
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, C23, D6, C2xC8, C22xC4, C4xS3, C22xS3, C22xC8, S3xC8, S3xC2xC4, S3xC2xC8

Smallest permutation representation of S3xC2xC8
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(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 28 35)(2 29 36)(3 30 37)(4 31 38)(5 32 39)(6 25 40)(7 26 33)(8 27 34)(9 46 17)(10 47 18)(11 48 19)(12 41 20)(13 42 21)(14 43 22)(15 44 23)(16 45 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)

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

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

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

S3xC2xC8 is a maximal subgroup of
D6:C16  D6.C42  C8:9D12  D6.4C42  C3:D4:C8  D6:C8:C2  D6:2M4(2)  D4:2S3:C4  D6:D8  D6:SD16  C4:C4.150D6  D6:2SD16  D6:1Q16  D12:C8  D6:3M4(2)  C42.30D6  (S3xC8):C4  C8:8D12  C8.27(C4xS3)  D6:2D8  D6:2Q16  C24:D4  D6:3D8  C24:14D4  D6:3Q16
S3xC2xC8 is a maximal quotient of
C42.282D6  Dic3.5M4(2)  C3:D4:C8  Dic6:C8  C42.200D6  D12:C8  D12.4C8  C16.12D6

48 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H6A6B6C8A···8H8I···8P12A12B12C12D24A···24H
order122222223444444446668···88···81212121224···24
size111133332111133332221···13···322222···2

48 irreducible representations

dim111111111222222
type++++++++
imageC1C2C2C2C2C4C4C4C8S3D6D6C4xS3C4xS3S3xC8
kernelS3xC2xC8S3xC8C2xC3:C8C2xC24S3xC2xC4C4xS3C2xDic3C22xS3D6C2xC8C8C2xC4C4C22C2
# reps1411142216121228

Matrix representation of S3xC2xC8 in GL3(F73) generated by

7200
0720
0072
,
7200
0220
0022
,
100
07272
010
,
7200
0720
011
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,22,0,0,0,22],[1,0,0,0,72,1,0,72,0],[72,0,0,0,72,1,0,0,1] >;

S3xC2xC8 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_8
% in TeX

G:=Group("S3xC2xC8");
// GroupNames label

G:=SmallGroup(96,106);
// by ID

G=gap.SmallGroup(96,106);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,50,69,2309]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<