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G = D6.D6  order 144 = 24·32

1st non-split extension by D6 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D6.5D6, C12.32D6, Dic3.6D6, C4.17S32, (C4×S3)⋊4S3, (S3×C12)⋊1C2, C3⋊D12⋊7C2, D6⋊S3⋊6C2, C3⋊2(C4○D12), C32⋊2Q8⋊6C2, C32⋊3(C4○D4), C6.5(C22×S3), (C3×C6).5C23, (S3×C6).6C22, (C3×C12).31C22, C3⋊Dic3.13C22, (C3×Dic3).6C22, C2.8(C2×S32), (C4×C3⋊S3)⋊6C2, (C2×C3⋊S3).13C22, SmallGroup(144,141)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for D6.D6
 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=c5 >

Subgroups: 288 in 88 conjugacy classes, 32 normal (12 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×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, D6.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, C2×S32, D6.D6

Character table of D6.D6

 class 12A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H12I12J
 size 11661822411661822466662222446666
ρ1111111111111111111111111111111    trivial
ρ211-111111-1-11-1-1111-1-111-1-1-1-1-1-1-1-111    linear of order 2
ρ31111-1111-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411-11-111111-11-1111-1-11111111111-1-1    linear of order 2
ρ511-1-1-1111-1-1111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ6111-1-1111111-1-111111-1-1111111-1-111    linear of order 2
ρ711-1-1111111-1-11111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ8111-11111-1-1-11-111111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ922-2002-1-1-2-2200-12-11100-2-2111100-1-1    orthogonal lifted from D6
ρ1022-2002-1-122-200-12-1110022-1-1-1-10011    orthogonal lifted from D6
ρ11222002-1-1-2-2-200-12-1-1-100-2-211110011    orthogonal lifted from D6
ρ12220-20-12-1220-202-1-10011-1-122-1-11100    orthogonal lifted from D6
ρ13220-20-12-1-2-20202-1-1001111-2-211-1-100    orthogonal lifted from D6
ρ1422020-12-1-2-20-202-1-100-1-111-2-2111100    orthogonal lifted from D6
ρ1522020-12-1220202-1-100-1-1-1-122-1-1-1-100    orthogonal lifted from S3
ρ16222002-1-122200-12-1-1-10022-1-1-1-100-1-1    orthogonal lifted from S3
ρ172-20002222i-2i000-2-2-20000-2i2i2i-2i2i-2i0000    complex lifted from C4○D4
ρ182-2000222-2i2i000-2-2-200002i-2i-2i2i-2i2i0000    complex lifted from C4○D4
ρ192-2000-12-12i-2i000-21100-√-3√-3i-i2i-2i-ii√3-√300    complex lifted from C4○D12
ρ202-2000-12-1-2i2i000-21100√-3-√-3-ii-2i2ii-i√3-√300    complex lifted from C4○D12
ρ212-20002-1-12i-2i0001-21√-3-√-300-2i2i-ii-ii00-√3√3    complex lifted from C4○D12
ρ222-2000-12-12i-2i000-21100√-3-√-3i-i2i-2i-ii-√3√300    complex lifted from C4○D12
ρ232-2000-12-1-2i2i000-21100-√-3√-3-ii-2i2ii-i-√3√300    complex lifted from C4○D12
ρ242-20002-1-1-2i2i0001-21√-3-√-3002i-2ii-ii-i00√3-√3    complex lifted from C4○D12
ρ252-20002-1-12i-2i0001-21-√-3√-300-2i2i-ii-ii00√3-√3    complex lifted from C4○D12
ρ262-20002-1-1-2i2i0001-21-√-3√-3002i-2ii-ii-i00-√3√3    complex lifted from C4○D12
ρ2744000-2-21-4-4000-2-2100002222-1-10000    orthogonal lifted from C2×S32
ρ2844000-2-2144000-2-210000-2-2-2-2110000    orthogonal lifted from S32
ρ294-4000-2-214i-4i00022-100002i-2i-2i2ii-i0000    complex faithful
ρ304-4000-2-21-4i4i00022-10000-2i2i2i-2i-ii0000    complex faithful

Permutation representations of D6.D6
►On 24 points - transitive group 24T228
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 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)

(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 18 7 24)(2 23 8 17)(3 16 9 22)(4 21 10 15)(5 14 11 20)(6 19 12 13)
 
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), (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18,7,24)(2,23,8,17)(3,16,9,22)(4,21,10,15)(5,14,11,20)(6,19,12,13)>;
 
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), (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18,7,24)(2,23,8,17)(3,16,9,22)(4,21,10,15)(5,14,11,20)(6,19,12,13) );
 
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)], [(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18,7,24),(2,23,8,17),(3,16,9,22),(4,21,10,15),(5,14,11,20),(6,19,12,13)]])
 
G:=TransitiveGroup(24,228);
 

D6.D6 is a maximal subgroup of
 C32⋊C4≀C2  C24.63D6  C24.64D6  C24.D6  C32⋊D8⋊5C2  C32⋊D8⋊C2  C32⋊Q16⋊C2  S3×C4○D12  D12⋊23D6  Dic6.24D6  D12⋊12D6  D12⋊13D6  D12.25D6  Dic6.26D6  D12⋊16D6  D6.D18  C12.91S32  (S3×C6).D6  D6.S32  Dic3.S32  C12.73S32  C12.95S32
D6.D6 is a maximal quotient of
 C62.20C23  D6⋊Dic6  Dic3.D12  C62.23C23  C62.25C23  C62.29C23  C62.32C23  C62.35C23  C62.37C23  C62.38C23  C62.40C23  C62.44C23  C4×D6⋊S3  C4×C3⋊D12  C62.75C23  D6⋊D12  C62.82C23  C62.85C23  C4×C32⋊2Q8  D6.D18  C12.91S32  (S3×C6).D6  D6.S32  Dic3.S32  C12.73S32  C12.95S32

Matrix representation of D6.D6 ►in GL4(𝔽5) generated by

0030
0101
3010
0400
,
0101
0030
0200
1020
,
3010
0303
1000
0200
,
0004
0020
0200
1000
G:=sub<GL(4,GF(5))| [0,0,3,0,0,1,0,4,3,0,1,0,0,1,0,0],[0,0,0,1,1,0,2,0,0,3,0,2,1,0,0,0],[3,0,1,0,0,3,0,2,1,0,0,0,0,3,0,0],[0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,0] >;
 

D6.D6 in GAP, Magma, Sage, TeX 

D_6.D_6
 
% in TeX
 
G:=Group("D6.D6");
 
// GroupNames label
 
G:=SmallGroup(144,141);
 
// by ID
 
G=gap.SmallGroup(144,141);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,50,490,3461]);
 
// 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=c^5>;
 
// generators/relations
 

Export

Character table of D6.D6 in TeX

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