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G = C8.6D6order 96 = 25·3

3rd non-split extension by C8 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.6D6, C33SD32, Q161S3, C6.10D8, C12.5D4, D24.2C2, C24.4C22, C3⋊C163C2, (C3×Q16)⋊1C2, C2.6(D4⋊S3), C4.3(C3⋊D4), SmallGroup(96,35)

Series: Derived Chief Lower central Upper central

C1C24 — C8.6D6
C1C3C6C12C24D24 — C8.6D6
C3C6C12C24 — C8.6D6
C1C2C4C8Q16

Generators and relations for C8.6D6
 G = < a,b,c | a8=1, b6=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b5 >

24C2
4C4
12C22
8S3
2Q8
6D4
4D6
4C12
3C16
3D8
2D12
2C3×Q8
3SD32

Character table of C8.6D6

 class 12A2B34A4B68A8B12A12B12C16A16B16C16D24A24B
 size 1124228222488666644
ρ1111111111111111111    trivial
ρ211-1111111111-1-1-1-111    linear of order 2
ρ311111-11111-1-1-1-1-1-111    linear of order 2
ρ411-111-11111-1-1111111    linear of order 2
ρ5220-122-122-1-1-10000-1-1    orthogonal lifted from S3
ρ6220-12-2-122-1110000-1-1    orthogonal lifted from D6
ρ72202202-2-22000000-2-2    orthogonal lifted from D4
ρ82202-20200-200-22-2200    orthogonal lifted from D8
ρ92202-20200-2002-22-200    orthogonal lifted from D8
ρ10220-120-1-2-2-1-3--3000011    complex lifted from C3⋊D4
ρ11220-120-1-2-2-1--3-3000011    complex lifted from C3⋊D4
ρ122-20200-22-2000ζ16131611ζ16716ζ165163ζ1615169-22    complex lifted from SD32
ρ132-20200-2-22000ζ16716ζ165163ζ1615169ζ161316112-2    complex lifted from SD32
ρ142-20200-2-22000ζ1615169ζ16131611ζ16716ζ1651632-2    complex lifted from SD32
ρ152-20200-22-2000ζ165163ζ1615169ζ16131611ζ16716-22    complex lifted from SD32
ρ16440-2-40-200200000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ174-40-200222-2200000002-2    orthogonal faithful, Schur index 2
ρ184-40-2002-22220000000-22    orthogonal faithful, Schur index 2

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

C8.6D6 is a maximal subgroup of
S3×SD32  D48⋊C2  Q32⋊S3  D485C2  C24.27C23  Q16⋊D6  Q16.D6  C9⋊SD32  D24.S3  C24.49D6  C3210SD32  C15⋊SD32  C24.D10  C8.6D30
C8.6D6 is a maximal quotient of
C6.SD32  C6.D16  C6.5Q32  C9⋊SD32  D24.S3  C24.49D6  C3210SD32  C15⋊SD32  C24.D10  C8.6D30

Matrix representation of C8.6D6 in GL4(𝔽7) generated by

6302
1611
2223
3426
,
2515
0024
2115
1214
,
0102
2663
2652
4243
G:=sub<GL(4,GF(7))| [6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,0,2,1,5,0,1,2,1,2,1,1,5,4,5,4],[0,2,2,4,1,6,6,2,0,6,5,4,2,3,2,3] >;

C8.6D6 in GAP, Magma, Sage, TeX

C_8._6D_6
% in TeX

G:=Group("C8.6D6");
// GroupNames label

G:=SmallGroup(96,35);
// by ID

G=gap.SmallGroup(96,35);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,103,218,116,122,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^6=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^5>;
// generators/relations

Export

Subgroup lattice of C8.6D6 in TeX
Character table of C8.6D6 in TeX

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