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G = C8.D6order 96 = 25·3

1st non-split extension by C8 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.1D6, C4.15D12, C12.13D4, Dic122C2, M4(2)⋊2S3, C24.1C22, C22.6D12, C12.33C23, D12.8C22, Dic6.8C22, (C2×C6).6D4, C24⋊C22C2, C6.14(C2×D4), (C2×C4).16D6, (C2×Dic6)⋊8C2, C4○D12.4C2, C2.16(C2×D12), C31(C8.C22), (C3×M4(2))⋊2C2, C4.31(C22×S3), (C2×C12).28C22, SmallGroup(96,116)

Series: Derived Chief Lower central Upper central

C1C12 — C8.D6
C1C3C6C12D12C4○D12 — C8.D6
C3C6C12 — C8.D6
C1C2C2×C4M4(2)

Generators and relations for C8.D6
 G = < a,b,c | a8=1, b6=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b5 >

Subgroups: 146 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], Dic3 [×3], C12 [×2], D6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C24 [×2], Dic6, Dic6 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, C24⋊C2 [×2], Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, C8.D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D12 [×2], C22×S3, C8.C22, C2×D12, C8.D6

Character table of C8.D6

 class 12A2B2C34A4B4C4D4E6A6B8A8B12A12B12C24A24B24C24D
 size 1121222212121224442244444
ρ1111111111111111111111    trivial
ρ211-111-11-11-11-1-1111-11-1-11    linear of order 2
ρ3111-1111-1-1-111111111111    linear of order 2
ρ411-111-111-1-11-11-111-1-111-1    linear of order 2
ρ5111-111111-111-1-1111-1-1-1-1    linear of order 2
ρ611-1-11-111-111-1-1111-11-1-11    linear of order 2
ρ71111111-1-1111-1-1111-1-1-1-1    linear of order 2
ρ811-1-11-11-1111-11-111-1-111-1    linear of order 2
ρ92220-122000-1-1-2-2-1-1-11111    orthogonal lifted from D6
ρ102220-122000-1-122-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122202-2-20002200-2-2-20000    orthogonal lifted from D4
ρ1222-20-1-22000-112-2-1-111-1-11    orthogonal lifted from D6
ρ1322-20-1-22000-11-22-1-11-111-1    orthogonal lifted from D6
ρ1422-2022-20002-200-2-220000    orthogonal lifted from D4
ρ152220-1-2-2000-1-100111-3-333    orthogonal lifted from D12
ρ162220-1-2-2000-1-10011133-3-3    orthogonal lifted from D12
ρ1722-20-12-2000-110011-1-33-33    orthogonal lifted from D12
ρ1822-20-12-2000-110011-13-33-3    orthogonal lifted from D12
ρ194-400400000-40000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-400-2000002000-232300000    symplectic faithful, Schur index 2
ρ214-400-200000200023-2300000    symplectic faithful, Schur index 2

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

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

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

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

C8.D6 is a maximal subgroup of
M4(2)⋊D6  D12.2D4  D12.4D4  D12.7D4  C425D6  Q8.14D12  C24.18D4  C24.42D4  C24.9C23  D4.11D12  D4.13D12  D84D6  D86D6  S3×C8.C22  SD16.D6  C8.D18  C24.3D6  Dic12⋊S3  D12.29D6  Dic6.29D6  C24.5D6  Dic60⋊C2  C24.2D10  C20.D12  D12.33D10  C8.D30
C8.D6 is a maximal quotient of
C8⋊Dic6  C42.14D6  C42.16D6  C42.20D6  C8.D12  Dic12⋊C4  C23.39D12  C23.15D12  D12.32D4  C22.D24  Dic614D4  Dic6.32D4  Dic6.3Q8  C12⋊SD16  C42.36D6  D124Q8  C4⋊Dic12  Dic64Q8  C23.51D12  C23.52D12  C23.54D12  C242D4  C24.4D4  C8.D18  C24.3D6  Dic12⋊S3  D12.29D6  Dic6.29D6  C24.5D6  Dic60⋊C2  C24.2D10  C20.D12  D12.33D10  C8.D30

Matrix representation of C8.D6 in GL6(𝔽73)

100000
010000
000010
00117271
000100
0010072
,
010000
7210000
0046000
0004600
0000270
004646027
,
7210000
010000
0046000
0002700
0027274619
00460027

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,72,0,0,0,0,0,71,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,46,0,0,46,0,0,0,46,0,46,0,0,0,0,27,0,0,0,0,0,0,27],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,46,0,27,46,0,0,0,27,27,0,0,0,0,0,46,0,0,0,0,0,19,27] >;

C8.D6 in GAP, Magma, Sage, TeX

C_8.D_6
% in TeX

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

G:=SmallGroup(96,116);
// by ID

G=gap.SmallGroup(96,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,579,69,2309]);
// Polycyclic

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

Export

Character table of C8.D6 in TeX

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