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G = D6.6D6  order 144 = 24·32

2nd non-split extension by D6 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D6.6D6, Dic6⋊5S3, C12.27D6, Dic3.9D6, C4.7S32, (C4×S3)⋊2S3, (S3×C12)⋊3C2, C12⋊S3⋊4C2, C3⋊D12⋊4C2, C3⋊1(C4○D12), (C3×Dic6)⋊7C2, C6.D6⋊1C2, C32⋊4(C4○D4), C6.6(C22×S3), (C3×C6).6C23, C3⋊1(Q8⋊3S3), (S3×C6).7C22, (C3×C12).20C22, (C3×Dic3).4C22, C2.9(C2×S32), (C2×C3⋊S3).4C22, SmallGroup(144,142)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — D6.6D6
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — C3⋊D12 — D6.6D6
Lower central
C32 — C3×C6 — D6.6D6
Upper central
C1 — C2 — C4

Generators and relations for D6.6D6
 G = < a,b,c,d | a6=b2=1, c6=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 316 in 88 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, Q8⋊3S3, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C12⋊S3, D6.6D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, Q8⋊3S3, C2×S32, D6.6D6

Character table of D6.6D6

 class 12A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E12A12B12C12D12E12F12G12H12I
 size 1161818224233662246622444661212
ρ1111111111111111111111111111    trivial
ρ21111-1111-1-1-11-111111-1-1-1-1-1-1-11-1    linear of order 2
ρ311-1-1-11111-1-111111-1-111111-1-111    linear of order 2
ρ411-1-11111-1111-1111-1-1-1-1-1-1-1111-1    linear of order 2
ρ5111-11111-1-1-1-1111111-1-1-1-1-1-1-1-11    linear of order 2
ρ6111-1-1111111-1-1111111111111-1-1    linear of order 2
ρ711-11-1111-111-11111-1-1-1-1-1-1-111-11    linear of order 2
ρ811-1111111-1-1-1-1111-1-111111-1-1-1-1    linear of order 2
ρ922000-12-1200-2-2-12-10022-1-1-10011    orthogonal lifted from D6
ρ1022000-12-1-2002-2-12-100-2-211100-11    orthogonal lifted from D6
ρ1122-2002-1-12-2-2002-1-111-1-1-12-11100    orthogonal lifted from D6
ρ12222002-1-1222002-1-1-1-1-1-1-12-1-1-100    orthogonal lifted from S3
ρ1322000-12-1-200-22-12-100-2-2111001-1    orthogonal lifted from D6
ρ1422-2002-1-1-222002-1-111111-21-1-100    orthogonal lifted from D6
ρ15222002-1-1-2-2-2002-1-1-1-1111-211100    orthogonal lifted from D6
ρ1622000-12-120022-12-10022-1-1-100-1-1    orthogonal lifted from S3
ρ172-20002220-2i2i00-2-2-20000000-2i2i00    complex lifted from C4○D4
ρ182-200022202i-2i00-2-2-200000002i-2i00    complex lifted from C4○D4
ρ192-20002-1-102i-2i00-211-√-3√-3-√3√3-√30√3-ii00    complex lifted from C4○D12
ρ202-20002-1-10-2i2i00-211√-3-√-3-√3√3-√30√3i-i00    complex lifted from C4○D12
ρ212-20002-1-102i-2i00-211√-3-√-3√3-√3√30-√3-ii00    complex lifted from C4○D12
ρ222-20002-1-10-2i2i00-211-√-3√-3√3-√3√30-√3i-i00    complex lifted from C4○D12
ρ2344000-2-21-40000-2-210022-12-10000    orthogonal lifted from C2×S32
ρ244-4000-24-2000002-4200000000000    orthogonal lifted from Q8⋊3S3, Schur index 2
ρ2544000-2-2140000-2-2100-2-21-210000    orthogonal lifted from S32
ρ264-4000-2-210000022-100-2√32√3√30-√30000    orthogonal faithful
ρ274-4000-2-210000022-1002√3-2√3-√30√30000    orthogonal faithful

Permutation representations of D6.6D6
►On 24 points - transitive group 24T222
Generators in S24
(1 3 5 7 9 11)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 24 22 20 18 16)

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

(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 7 21)(2 14 8 20)(3 13 9 19)(4 24 10 18)(5 23 11 17)(6 22 12 16)
 
G:=sub<Sym(24)| (1,3,5,7,9,11)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15,7,21)(2,14,8,20)(3,13,9,19)(4,24,10,18)(5,23,11,17)(6,22,12,16)>;
 
G:=Group( (1,3,5,7,9,11)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15,7,21)(2,14,8,20)(3,13,9,19)(4,24,10,18)(5,23,11,17)(6,22,12,16) );
 
G=PermutationGroup([[(1,3,5,7,9,11),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,24,22,20,18,16)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,13),(12,14)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15,7,21),(2,14,8,20),(3,13,9,19),(4,24,10,18),(5,23,11,17),(6,22,12,16)]])
 
G:=TransitiveGroup(24,222);
 

D6.6D6 is a maximal subgroup of
 C24⋊1D6  Dic12⋊S3  D6.1D12  D6.3D12  D12.7D6  Dic6.20D6  Dic6.22D6  D12.13D6  D12.33D6  S3×C4○D12  D12⋊27D6  Dic6⋊12D6  D12⋊13D6  Dic6.26D6  S3×Q8⋊3S3  Dic6⋊5D9  Dic9.D6  C12⋊S3⋊S3  C12.84S32  D6.3S32  Dic3.S32  C12.40S32  C12.58S32  C12⋊S3⋊12S3
D6.6D6 is a maximal quotient of
 C62.6C23  C62.11C23  Dic3×Dic6  Dic3.Dic6  C62.18C23  C62.20C23  C62.24C23  D6⋊6Dic6  C62.31C23  C12.28D12  C62.38C23  C62.39C23  C62.58C23  Dic3⋊5D12  C62.67C23  C62.74C23  C62.77C23  C12⋊7D12  Dic3⋊3D12  Dic6⋊5D9  Dic9.D6  C12.S32  D6.3S32  Dic3.S32  C12.40S32  C12.58S32  C12⋊S3⋊12S3

Matrix representation of D6.6D6 ►in GL6(𝔽13)

1200000
0120000
0012100
0012000
000010
000001
,
080000
500000
001000
0011200
000010
000001
,
010000
1200000
0012000
0001200
0000121
0000120
,
500000
080000
001000
000100
0000120
0000121

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;
 

D6.6D6 in GAP, Magma, Sage, TeX 

D_6._6D_6
 
% in TeX
 
G:=Group("D6.6D6");
 
// GroupNames label
 
G:=SmallGroup(144,142);
 
// by ID
 
G=gap.SmallGroup(144,142);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,55,116,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=1,c^6=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
 
// generators/relations
 

Export

Character table of D6.6D6 in TeX

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