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G = C6xD8order 96 = 25·3

Direct product of C6 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C6xD8, C12.41D4, C24:10C22, C12.44C23, C8:2(C2xC6), (C2xC8):3C6, (C2xC24):8C2, (C2xD4):4C6, D4:1(C2xC6), C4.6(C3xD4), (C6xD4):13C2, C6.74(C2xD4), C2.11(C6xD4), (C2xC6).52D4, C4.1(C22xC6), (C3xD4):10C22, C22.14(C3xD4), (C2xC12).129C22, (C2xC4).25(C2xC6), SmallGroup(96,179)

Series: Derived Chief Lower central Upper central

C1C4 — C6xD8
C1C2C4C12C3xD4C3xD8 — C6xD8
C1C2C4 — C6xD8
C1C2xC6C2xC12 — C6xD8

Generators and relations for C6xD8
 G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2xC4, D4, D4, C23, C12, C2xC6, C2xC6, C2xC8, D8, C2xD4, C24, C2xC12, C3xD4, C3xD4, C22xC6, C2xD8, C2xC24, C3xD8, C6xD4, C6xD8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, D8, C2xD4, C3xD4, C22xC6, C2xD8, C3xD8, C6xD4, C6xD8

Smallest permutation representation of C6xD8
On 48 points
Generators in S48
(1 14 39 22 29 45)(2 15 40 23 30 46)(3 16 33 24 31 47)(4 9 34 17 32 48)(5 10 35 18 25 41)(6 11 36 19 26 42)(7 12 37 20 27 43)(8 13 38 21 28 44)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)

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

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

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

C6xD8 is a maximal subgroup of
D8:1Dic3  D8.Dic3  D8.D6  Dic3:D8  C24:5D4  D8:Dic3  (C6xD8).C2  C24:11D4  C24.22D4  D12:D4  D6:3D8  Dic6:D4  C24:12D4  C24.23D4  D8:13D6

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B6A···6F6G···6N8A8B8C8D12A12B12C12D24A···24H
order1222222233446···66···688881212121224···24
size1111444411221···14···4222222222···2

42 irreducible representations

dim11111111222222
type+++++++
imageC1C2C2C2C3C6C6C6D4D4D8C3xD4C3xD4C3xD8
kernelC6xD8C2xC24C3xD8C6xD4C2xD8C2xC8D8C2xD4C12C2xC6C6C4C22C2
# reps11422284114228

Matrix representation of C6xD8 in GL3(F73) generated by

7200
0640
0064
,
7200
04157
0320
,
100
0057
0410
G:=sub<GL(3,GF(73))| [72,0,0,0,64,0,0,0,64],[72,0,0,0,41,32,0,57,0],[1,0,0,0,0,41,0,57,0] >;

C6xD8 in GAP, Magma, Sage, TeX

C_6\times D_8
% in TeX

G:=Group("C6xD8");
// GroupNames label

G:=SmallGroup(96,179);
// by ID

G=gap.SmallGroup(96,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,2164,1090,88]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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