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G = D12⋊1D4  order 192 = 26·3

1st semidirect product of D12 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊1D4, Dic6⋊1D4, M4(2)⋊2D6, C23.7D12, C8⋊D6⋊5C2, C4.79(S3×D4), D4⋊6D6⋊2C2, C12⋊3D4⋊1C2, C3⋊1(D4⋊4D4), (C2×D4).14D6, C4.D4⋊2S3, C12.92(C2×D4), D12⋊C4⋊2C2, C6.16C22≀C2, (C2×C12).4C23, (C22×C6).20D4, (C2×D12)⋊11C22, C4○D12.2C22, (C4×Dic3)⋊2C22, (C6×D4).14C22, C22.11(C2×D12), C2.19(D6⋊D4), (C3×M4(2))⋊9C22, (C2×C6).21(C2×D4), (C3×C4.D4)⋊4C2, (C2×C4).4(C22×S3), SmallGroup(192,306)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for D12⋊1D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a7b, dcd=c-1 >

Subgroups: 656 in 168 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4.D4, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, C24⋊C2, D24, C4×Dic3, C3×M4(2), C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, D4⋊4D4, D12⋊C4, C3×C4.D4, C8⋊D6, C12⋊3D4, D4⋊6D6, D12⋊1D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4⋊4D4, D6⋊D4, D12⋊1D4

Character table of D12⋊1D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D8A8B12A12B24A24B24C24D
 size 1124412122422212121212248888448888
ρ1111111111111111111111111111    trivial
ρ21111111-1111-111-11111-1-111-1-1-1-1    linear of order 2
ρ3111-1-1-11-111111-1111-1-11-111-11-11    linear of order 2
ρ4111-1-1-111111-11-1-111-1-1-11111-11-1    linear of order 2
ρ511111-1-111111-1-111111-1-111-1-1-1-1    linear of order 2
ρ611111-1-1-1111-1-1-1-1111111111111    linear of order 2
ρ7111-1-11-1-11111-11111-1-1-11111-11-1    linear of order 2
ρ8111-1-11-11111-1-11-111-1-11-111-11-11    linear of order 2
ρ922-2002002-2200-202-200002-20000    orthogonal lifted from D4
ρ10222-2-2000-1220000-1-1112-2-1-11-11-1    orthogonal lifted from D6
ρ1122222000-1220000-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ12222-2-2000-1220000-1-111-22-1-1-11-11    orthogonal lifted from D6
ρ13222-220002-2-20000222-200-2-20000    orthogonal lifted from D4
ρ1422222000-1220000-1-1-1-1-2-2-1-11111    orthogonal lifted from D6
ρ152222-20002-2-2000022-2200-2-20000    orthogonal lifted from D4
ρ1622-200-2002-2200202-200002-20000    orthogonal lifted from D4
ρ1722-20002022-20-2002-20000-220000    orthogonal lifted from D4
ρ1822-2000-2022-202002-20000-220000    orthogonal lifted from D4
ρ192222-2000-1-2-20000-1-11-10011√3-√3-√3√3    orthogonal lifted from D12
ρ20222-22000-1-2-20000-1-1-110011√3√3-√3-√3    orthogonal lifted from D12
ρ21222-22000-1-2-20000-1-1-110011-√3-√3√3√3    orthogonal lifted from D12
ρ222222-2000-1-2-20000-1-11-10011-√3√3√3-√3    orthogonal lifted from D12
ρ234-4000000400-2002-400000000000    orthogonal lifted from D4⋊4D4
ρ2444-400000-24-40000-2200002-20000    orthogonal lifted from S3×D4
ρ2544-400000-2-440000-220000-220000    orthogonal lifted from S3×D4
ρ264-4000000400200-2-400000000000    orthogonal lifted from D4⋊4D4
ρ278-8000000-4000000400000000000    orthogonal faithful

Permutation representations of D12⋊1D4
►On 24 points - transitive group 24T343
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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)

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

(2 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)
 
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (2,6)(3,11)(5,9)(8,12)(13,22,19,16)(14,15,20,21)(17,18,23,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)>;
 
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), (1,15)(2,14)(3,13)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16), (2,6)(3,11)(5,9)(8,12)(13,22,19,16)(14,15,20,21)(17,18,23,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24) );
 
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)], [(1,15),(2,14),(3,13),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16)], [(2,6),(3,11),(5,9),(8,12),(13,22,19,16),(14,15,20,21),(17,18,23,24)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24)]])
 
G:=TransitiveGroup(24,343);
 

Matrix representation of D12⋊1D4 ►in GL6(𝔽73)

110000
7200000
00722500
0070100
00111701
00620720
,
6670000
1470000
007242710
00052703
000491163
000246311
,
7200000
110000
0014800
0037200
006249720
001049072
,
7200000
110000
0072000
0070100
00116601
0063710

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,70,11,62,0,0,25,1,17,0,0,0,0,0,0,72,0,0,0,0,1,0],[66,14,0,0,0,0,7,7,0,0,0,0,0,0,72,0,0,0,0,0,42,52,49,24,0,0,71,70,11,63,0,0,0,3,63,11],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,3,62,10,0,0,48,72,49,49,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,70,11,63,0,0,0,1,66,7,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

D12⋊1D4 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_1D_4
 
% in TeX
 
G:=Group("D12:1D4");
 
// GroupNames label
 
G:=SmallGroup(192,306);
 
// by ID
 
G=gap.SmallGroup(192,306);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,226,1123,570,136,1684,438,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^7*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D12⋊1D4 in TeX

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