Copied to
clipboard

G = C8⋊D6order 96 = 25·3

1st semidirect product of C8 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C81D6, D242C2, C241C22, C12.12D4, C4.14D12, D124C22, M4(2)⋊1S3, C22.5D12, C12.32C23, Dic64C22, (C2×C6).5D4, C24⋊C21C2, C4○D122C2, (C2×D12)⋊7C2, C31(C8⋊C22), C6.13(C2×D4), (C2×C4).15D6, C2.15(C2×D12), (C3×M4(2))⋊1C2, C4.30(C22×S3), (C2×C12).27C22, SmallGroup(96,115)

Series: Derived Chief Lower central Upper central

C1C12 — C8⋊D6
C1C3C6C12D12C2×D12 — C8⋊D6
C3C6C12 — C8⋊D6
C1C2C2×C4M4(2)

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

Subgroups: 210 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, Dic3, C12 [×2], D6 [×5], C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C24 [×2], Dic6, C4×S3, D12, D12 [×2], D12, C3⋊D4, C2×C12, C22×S3, C8⋊C22, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, 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 12A2B2C2D2E34A4B4C6A6B8A8B12A12B12C24A24B24C24D
 size 1121212122221224442244444
ρ1111111111111111111111    trivial
ρ21111-1-1111111-1-1111-1-1-1-1    linear of order 2
ρ311-11-111-11-11-11-111-11-1-11    linear of order 2
ρ4111-111111-111-1-1111-1-1-1-1    linear of order 2
ρ511-1-11-11-1111-11-111-11-1-11    linear of order 2
ρ611-111-11-11-11-1-1111-1-111-1    linear of order 2
ρ7111-1-1-1111-111111111111    linear of order 2
ρ811-1-1-111-1111-1-1111-1-111-1    linear of order 2
ρ922-200022-202-200-2-220000    orthogonal lifted from D4
ρ10222000-1220-1-1-2-2-1-1-11111    orthogonal lifted from D6
ρ11222000-1220-1-122-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-2000-1-220-11-22-1-111-1-11    orthogonal lifted from D6
ρ132220002-2-202200-2-2-20000    orthogonal lifted from D4
ρ1422-2000-1-220-112-2-1-11-111-1    orthogonal lifted from D6
ρ1522-2000-12-20-110011-1-33-33    orthogonal lifted from D12
ρ1622-2000-12-20-110011-13-33-3    orthogonal lifted from D12
ρ17222000-1-2-20-1-100111-3-333    orthogonal lifted from D12
ρ18222000-1-2-20-1-10011133-3-3    orthogonal lifted from D12
ρ194-400004000-40000000000    orthogonal lifted from C8⋊C22
ρ204-40000-20002000-232300000    orthogonal faithful
ρ214-40000-2000200023-2300000    orthogonal faithful

Permutation representations of C8⋊D6
On 24 points - transitive group 24T107
Generators in S24
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 13 21)(2 10 22 6 14 18)(3 15 23)(4 12 24 8 16 20)(5 9 17)(7 11 19)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(10 16)(11 15)(12 14)

G:=sub<Sym(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), (1,13,21)(2,10,22,6,14,18)(3,15,23)(4,12,24,8,16,20)(5,9,17)(7,11,19), (1,21)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(10,16)(11,15)(12,14)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,13,21)(2,10,22,6,14,18)(3,15,23)(4,12,24,8,16,20)(5,9,17)(7,11,19), (1,21)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(10,16)(11,15)(12,14) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,13,21),(2,10,22,6,14,18),(3,15,23),(4,12,24,8,16,20),(5,9,17),(7,11,19)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,24),(7,23),(8,22),(10,16),(11,15),(12,14)])

G:=TransitiveGroup(24,107);

C8⋊D6 is a maximal subgroup of
D121D4  D12.3D4  D12.5D4  D12.6D4  Q85D12  D4.10D12  C24.19D4  C24.42D4  C24.9C23  D4.11D12  D4.12D12  S3×C8⋊C22  D85D6  D24⋊C22  C24.C23  C8⋊D18  C241D6  D24⋊S3  D1218D6  D12.28D6  C243D6  C24⋊D10  D24⋊D5  D6036C22  C60.38D4  C8⋊D30
C8⋊D6 is a maximal quotient of
C8⋊Dic6  C42.16D6  D24⋊C4  C8⋊D12  C42.19D6  C42.20D6  C23.40D12  D12.31D4  D1213D4  D1214D4  C23.43D12  C23.18D12  D123Q8  C4⋊D24  D12.19D4  D12.3Q8  Dic68D4  Dic63Q8  C23.52D12  C23.53D12  C23.54D12  C242D4  C243D4  C8⋊D18  C241D6  D24⋊S3  D1218D6  D12.28D6  C243D6  C24⋊D10  D24⋊D5  D6036C22  C60.38D4  C8⋊D30

Matrix representation of C8⋊D6 in GL4(𝔽73) generated by

0010
0001
71400
596600
,
1100
72000
007272
0010
,
727200
0100
00667
00147
G:=sub<GL(4,GF(73))| [0,0,7,59,0,0,14,66,1,0,0,0,0,1,0,0],[1,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,72,1,0,0,0,0,66,14,0,0,7,7] >;

C8⋊D6 in GAP, Magma, Sage, TeX

C_8\rtimes D_6
% in TeX

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

G:=SmallGroup(96,115);
// by ID

G=gap.SmallGroup(96,115);
# by ID

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

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

Export

Character table of C8⋊D6 in TeX

׿
×
𝔽