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G = C42⋊8D6  order 192 = 26·3

6th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊5D4, C42⋊8D6, Dic6⋊5D4, (C2×D4)⋊2D6, C4⋊1D4⋊5S3, C4.55(S3×D4), D4⋊6D6⋊4C2, C3⋊3(D4⋊4D4), C12.35(C2×D4), (C6×D4)⋊2C22, (C4×C12)⋊14C22, C6.53C22≀C2, D12⋊6C22⋊3C2, C12.D4⋊6C2, C42⋊4S3⋊13C2, (C22×C6).23D4, C4.Dic3⋊7C22, C2.21(C23⋊2D6), (C2×C12).395C23, C4○D12.21C22, C23.11(C3⋊D4), (C3×C4⋊1D4)⋊4C2, (C2×C6).526(C2×D4), C22.33(C2×C3⋊D4), (C2×C4).118(C22×S3), SmallGroup(192,636)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — C42⋊8D6
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D4⋊6D6 — C42⋊8D6
Lower central
C3 — C6 — C2×C12 — C42⋊8D6
Upper central
C1 — C2 — C2×C4 — C4⋊1D4

Generators and relations for C42⋊8D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=a-1b-1, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 560 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4.D4, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, C4.Dic3, D4⋊S3, D4.S3, C4×C12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, C6×D4, D4⋊4D4, C42⋊4S3, C12.D4, D12⋊6C22, C3×C4⋊1D4, D4⋊6D6, C42⋊8D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4⋊4D4, C23⋊2D6, C42⋊8D6

Character table of C42⋊8D6

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G8A8B12A12B12C12D12E12F
 size 112448121222244121222288882424444444
ρ1111111111111111111111111111111    trivial
ρ2111-1-11-11111-1-11-1111-111-1-111-1-1-11-1    linear of order 2
ρ311111-1-1-1111-1-1-1-11111-1-11111-1-1-11-1    linear of order 2
ρ4111-1-1-11-111111-11111-1-1-1-1-11111111    linear of order 2
ρ5111111-1-111111-1-11111111-1-1111111    linear of order 2
ρ6111-1-111-1111-1-1-11111-111-11-11-1-1-11-1    linear of order 2
ρ711111-111111-1-1111111-1-11-1-11-1-1-11-1    linear of order 2
ρ8111-1-1-1-11111111-1111-1-1-1-11-1111111    linear of order 2
ρ922-20000222-200-20-22-2000000-200020    orthogonal lifted from D4
ρ102222-20002-2-20000222-200200-2000-20    orthogonal lifted from D4
ρ1122-20000-222-20020-22-2000000-200020    orthogonal lifted from D4
ρ1222-2000-202-220002-22-20000002000-20    orthogonal lifted from D4
ρ1322222200-1222200-1-1-1-1-1-1-100-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422-2000202-22000-2-22-20000002000-20    orthogonal lifted from D4
ρ1522222-200-122-2-200-1-1-1-111-100-1111-11    orthogonal lifted from D6
ρ16222-220002-2-20000222200-200-2000-20    orthogonal lifted from D4
ρ17222-2-2-200-1222200-1-1-1111100-1-1-1-1-1-1    orthogonal lifted from D6
ρ18222-2-2200-122-2-200-1-1-11-1-1100-1111-11    orthogonal lifted from D6
ρ192222-2000-1-2-20000-1-1-11-√-3√-3-1001-√-3√-3√-31-√-3    complex lifted from C3⋊D4
ρ202222-2000-1-2-20000-1-1-11√-3-√-3-1001√-3-√-3-√-31√-3    complex lifted from C3⋊D4
ρ21222-22000-1-2-20000-1-1-1-1-√-3√-31001√-3-√-3-√-31√-3    complex lifted from C3⋊D4
ρ22222-22000-1-2-20000-1-1-1-1√-3-√-31001-√-3√-3√-31-√-3    complex lifted from C3⋊D4
ρ234-40000004002-2000-4000000002-220-2    orthogonal lifted from D4⋊4D4
ρ2444-400000-2-4400002-22000000-200020    orthogonal lifted from S3×D4
ρ2544-400000-24-400002-220000002000-20    orthogonal lifted from S3×D4
ρ264-4000000400-22000-400000000-22-202    orthogonal lifted from D4⋊4D4
ρ274-4000000-2002-2002√-32-2√-30000000-1-√-31-√-3-1+√-301+√-3    complex faithful
ρ284-4000000-2002-200-2√-322√-30000000-1+√-31+√-3-1-√-301-√-3    complex faithful
ρ294-4000000-200-22002√-32-2√-300000001+√-3-1+√-31-√-30-1-√-3    complex faithful
ρ304-4000000-200-2200-2√-322√-300000001-√-3-1-√-31+√-30-1+√-3    complex faithful

Permutation representations of C42⋊8D6
►On 24 points - transitive group 24T355
Generators in S24
(1 14 17 4)(2 5 18 15)(3 16 13 6)

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

(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 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)
 
G:=sub<Sym(24)| (1,14,17,4)(2,5,18,15)(3,16,13,6), (1,4,17,14)(2,15,18,5)(3,6,13,16)(7,10,23,20)(8,21,24,11)(9,12,19,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24)>;
 
G:=Group( (1,14,17,4)(2,5,18,15)(3,16,13,6), (1,4,17,14)(2,15,18,5)(3,6,13,16)(7,10,23,20)(8,21,24,11)(9,12,19,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,23)(14,22)(15,21)(16,20)(17,19)(18,24) );
 
G=PermutationGroup([[(1,14,17,4),(2,5,18,15),(3,16,13,6)], [(1,4,17,14),(2,15,18,5),(3,6,13,16),(7,10,23,20),(8,21,24,11),(9,12,19,22)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,23),(14,22),(15,21),(16,20),(17,19),(18,24)]])
 
G:=TransitiveGroup(24,355);
 

Matrix representation of C42⋊8D6 ►in GL4(𝔽7) generated by

2301
1015
4446
0006
,
4164
2644
4446
5210
,
0410
1436
6662
0004
,
4632
5161
1633
4436
G:=sub<GL(4,GF(7))| [2,1,4,0,3,0,4,0,0,1,4,0,1,5,6,6],[4,2,4,5,1,6,4,2,6,4,4,1,4,4,6,0],[0,1,6,0,4,4,6,0,1,3,6,0,0,6,2,4],[4,5,1,4,6,1,6,4,3,6,3,3,2,1,3,6] >;
 

C42⋊8D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_8D_6
 
% in TeX
 
G:=Group("C4^2:8D6");
 
// GroupNames label
 
G:=SmallGroup(192,636);
 
// by ID
 
G=gap.SmallGroup(192,636);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,1123,570,297,136,1684,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C42⋊8D6 in TeX

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