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G = D12⋊6C22  order 96 = 25·3

4th semidirect product of D12 and C22 acting via C22/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.6D6, C12.15D4, D12⋊6C22, C12.12C23, Dic6⋊5C22, D4⋊S3⋊5C2, (C2×D4)⋊2S3, (C6×D4)⋊2C2, C3⋊C8⋊3C22, C4○D12⋊3C2, C3⋊4(C8⋊C22), D4.S3⋊5C2, (C2×C4).17D6, (C2×C6).39D4, C6.45(C2×D4), C4.Dic3⋊6C2, C4.16(C3⋊D4), C4.12(C22×S3), (C3×D4).6C22, (C2×C12).30C22, C22.10(C3⋊D4), C2.9(C2×C3⋊D4), SmallGroup(96,139)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D12⋊6C22
Chief series
C1 — C3 — C6 — C12 — D12 — C4○D12 — D12⋊6C22
Lower central
C3 — C6 — C12 — D12⋊6C22
Upper central
C1 — C2 — C2×C4 — C2×D4

Generators and relations for D12⋊6C22
 G = < a,b,c,d | a12=b2=c2=d2=1, bab=a-1, ac=ca, dad=a7, cbc=a6b, dbd=a3b, cd=dc >

Subgroups: 162 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C22, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, D12⋊6C22
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, C2×C3⋊D4, D12⋊6C22

Character table of D12⋊6C22

 class 12A2B2C2D2E34A4B4C6A6B6C6D6E6F6G8A8B12A12B
 size 1124412222122224444121244
ρ1111111111111111111111    trivial
ρ211-1-11-11-111-1-11-1-1111-11-1    linear of order 2
ρ3111-1-1-1111-1111-1-1-1-11111    linear of order 2
ρ411111-1111-11111111-1-111    linear of order 2
ρ511-11-1-11-111-1-1111-1-1-111-1    linear of order 2
ρ611-11-111-11-1-1-1111-1-11-11-1    linear of order 2
ρ711-1-1111-11-1-1-11-1-111-111-1    linear of order 2
ρ8111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ9222220-1220-1-1-1-1-1-1-100-1-1    orthogonal lifted from S3
ρ10222-2-20-1220-1-1-1111100-1-1    orthogonal lifted from D6
ρ1122-22-20-1-22011-1-1-11100-11    orthogonal lifted from D6
ρ1222-200022-20-2-22000000-22    orthogonal lifted from D4
ρ132220002-2-20222000000-2-2    orthogonal lifted from D4
ρ1422-2-220-1-22011-111-1-100-11    orthogonal lifted from D6
ρ15222000-1-2-20-1-1-1-√-3√-3-√-3√-30011    complex lifted from C3⋊D4
ρ16222000-1-2-20-1-1-1√-3-√-3√-3-√-30011    complex lifted from C3⋊D4
ρ1722-2000-12-2011-1√-3-√-3-√-3√-3001-1    complex lifted from C3⋊D4
ρ1822-2000-12-2011-1-√-3√-3√-3-√-3001-1    complex lifted from C3⋊D4
ρ194-40000400000-400000000    orthogonal lifted from C8⋊C22
ρ204-40000-2000-2√-32√-3200000000    complex faithful
ρ214-40000-20002√-3-2√-3200000000    complex faithful

Permutation representations of D12⋊6C22
►On 24 points - transitive group 24T118
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(2 8)(4 10)(6 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)
 
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(4,10)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)>;
 
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(4,10)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23) );
 
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(2,8),(4,10),(6,12),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23)]])
 
G:=TransitiveGroup(24,118);
 

D12⋊6C22 is a maximal subgroup of
 D12.2D4  D12.3D4  D12.14D4  C42⋊8D6  C24.23D4  C24.44D4  D12⋊18D4  D12.38D4  D8⋊13D6  SD16⋊13D6  S3×C8⋊C22  D8⋊4D6  C12.C24  D12.32C23  D12.33C23  D36⋊6C22  D12⋊20D6  D12.28D6  D12⋊9D6  D12.7D6  C62.131D4  C60.36D4  D60⋊30C22  D12⋊10D10  D20.9D6  D4.D30
D12⋊6C22 is a maximal quotient of
 C4⋊C4.225D6  C4○D12⋊C4  C4⋊C4.228D6  C4⋊C4.230D6  D4.3Dic6  C42.48D6  D4.1D12  C42.51D6  C6.Q16⋊C2  D12⋊17D4  C4⋊D4⋊S3  C3⋊C8⋊5D4  C42.72D6  C12⋊2D8  C42.74D6  Dic6⋊9D4  C42.76D6  D12⋊5Q8  C42.82D6  Dic6⋊5Q8  (C6×D4)⋊6C4  (C2×C6)⋊8D8  (C3×D4).31D4  D36⋊6C22  D12⋊20D6  D12.28D6  D12⋊9D6  D12.7D6  C62.131D4  C60.36D4  D60⋊30C22  D12⋊10D10  D20.9D6  D4.D30

Matrix representation of D12⋊6C22 ►in GL4(𝔽7) generated by

6463
2026
1115
1630
,
6000
0532
4355
1165
,
0145
1035
0010
0006
,
0660
6060
0010
0006
G:=sub<GL(4,GF(7))| [6,2,1,1,4,0,1,6,6,2,1,3,3,6,5,0],[6,0,4,1,0,5,3,1,0,3,5,6,0,2,5,5],[0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[0,6,0,0,6,0,0,0,6,6,1,0,0,0,0,6] >;
 

D12⋊6C22 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_6C_2^2
 
% in TeX
 
G:=Group("D12:6C2^2");
 
// GroupNames label
 
G:=SmallGroup(96,139);
 
// by ID
 
G=gap.SmallGroup(96,139);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,579,159,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c=a^6*b,d*b*d=a^3*b,c*d=d*c>;
 
// generators/relations
 

Export

Character table of D12⋊6C22 in TeX

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