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G = D6.C8order 96 = 25·3

The non-split extension by D6 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D6.C8, C486C2, C163S3, C8.20D6, Dic3.C8, C31M5(2), C24.24C22, C3⋊C164C2, C3⋊C8.2C4, C2.3(S3×C8), C6.2(C2×C8), (S3×C8).2C2, (C4×S3).2C4, C4.17(C4×S3), C12.22(C2×C4), SmallGroup(96,5)

Series: Derived Chief Lower central Upper central

C1C6 — D6.C8
C1C3C6C12C24S3×C8 — D6.C8
C3C6 — D6.C8
C1C8C16

Generators and relations for D6.C8
 G = < a,b,c | a6=b2=1, c8=a3, bab=a-1, ac=ca, cbc-1=a3b >

6C2
3C4
3C22
2S3
3C2×C4
3C8
3C2×C8
3C16
3M5(2)

Smallest permutation representation of D6.C8
On 48 points
Generators in S48
(1 20 47 9 28 39)(2 21 48 10 29 40)(3 22 33 11 30 41)(4 23 34 12 31 42)(5 24 35 13 32 43)(6 25 36 14 17 44)(7 26 37 15 18 45)(8 27 38 16 19 46)
(1 39)(2 48)(3 41)(4 34)(5 43)(6 36)(7 45)(8 38)(9 47)(10 40)(11 33)(12 42)(13 35)(14 44)(15 37)(16 46)(18 26)(20 28)(22 30)(24 32)
(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)

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

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

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

D6.C8 is a maximal subgroup of
D12.4C8  S3×M5(2)  C16.12D6  D8⋊D6  D48⋊C2  SD32⋊S3  Q32⋊S3  C16⋊D9  C24.61D6  C24.62D6  C48⋊S3  C40.52D6  D30.5C8  C80⋊S3  C15⋊M5(2)  D30.C8
D6.C8 is a maximal quotient of
Dic3⋊C16  C4810C4  D6⋊C16  C16⋊D9  C24.61D6  C24.62D6  C48⋊S3  C40.52D6  D30.5C8  C80⋊S3  C15⋊M5(2)  D30.C8

36 conjugacy classes

class 1 2A2B 3 4A4B4C 6 8A8B8C8D8E8F12A12B16A16B16C16D16E16F16G16H24A24B24C24D48A···48H
order12234446888888121216161616161616162424242448···48
size11621162111166222222666622222···2

36 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C4C4C8C8S3D6C4×S3M5(2)S3×C8D6.C8
kernelD6.C8C3⋊C16C48S3×C8C3⋊C8C4×S3Dic3D6C16C8C4C3C2C1
# reps11112244112448

Matrix representation of D6.C8 in GL2(𝔽41) generated by

4038
12
,
400
11
,
36
3938
G:=sub<GL(2,GF(41))| [40,1,38,2],[40,1,0,1],[3,39,6,38] >;

D6.C8 in GAP, Magma, Sage, TeX

D_6.C_8
% in TeX

G:=Group("D6.C8");
// GroupNames label

G:=SmallGroup(96,5);
// by ID

G=gap.SmallGroup(96,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,31,50,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^6=b^2=1,c^8=a^3,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations

Export

Subgroup lattice of D6.C8 in TeX

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