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G = D6.6D6order 144 = 24·32

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

metabelian, supersoluble, monomial

Aliases: D6.6D6, Dic65S3, C12.27D6, Dic3.9D6, C4.7S32, (C4×S3)⋊2S3, (S3×C12)⋊3C2, C12⋊S34C2, C3⋊D124C2, C31(C4○D12), (C3×Dic6)⋊7C2, C6.D61C2, C324(C4○D4), C6.6(C22×S3), (C3×C6).6C23, C31(Q83S3), (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

C1C3×C6 — D6.6D6
C1C3C32C3×C6S3×C6C3⋊D12 — D6.6D6
C32C3×C6 — D6.6D6
C1C2C4

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 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×7], C6 [×2], C6 [×2], C2×C4 [×3], D4 [×3], Q8, C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, D6 [×6], C2×C6, C4○D4, C3×S3, C3⋊S3 [×2], C3×C6, Dic6, C4×S3, C4×S3 [×4], D12 [×5], C3⋊D4 [×2], C2×C12, C3×Q8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C2×C3⋊S3 [×2], C4○D12, Q83S3, C6.D6 [×2], C3⋊D12 [×2], C3×Dic6, S3×C12, C12⋊S3, D6.6D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, C4○D12, Q83S3, 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--3-3-33-303-ii00    complex lifted from C4○D12
ρ202-20002-1-10-2i2i00-211-3--3-33-303i-i00    complex lifted from C4○D12
ρ212-20002-1-102i-2i00-211-3--33-330-3-ii00    complex lifted from C4○D12
ρ222-20002-1-10-2i2i00-211--3-33-330-3i-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 Q83S3, Schur index 2
ρ2544000-2-2140000-2-2100-2-21-210000    orthogonal lifted from S32
ρ264-4000-2-210000022-100-232330-30000    orthogonal faithful
ρ274-4000-2-210000022-10023-23-3030000    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 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(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 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)

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

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

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

G:=TransitiveGroup(24,222);

D6.6D6 is a maximal subgroup of
C241D6  Dic12⋊S3  D6.1D12  D6.3D12  D12.7D6  Dic6.20D6  Dic6.22D6  D12.13D6  D12.33D6  S3×C4○D12  D1227D6  Dic612D6  D1213D6  Dic6.26D6  S3×Q83S3  Dic65D9  Dic9.D6  C12⋊S3⋊S3  C12.84S32  D6.3S32  Dic3.S32  C12.40S32  C12.58S32  C12⋊S312S3
D6.6D6 is a maximal quotient of
C62.6C23  C62.11C23  Dic3×Dic6  Dic3.Dic6  C62.18C23  C62.20C23  C62.24C23  D66Dic6  C62.31C23  C12.28D12  C62.38C23  C62.39C23  C62.58C23  Dic35D12  C62.67C23  C62.74C23  C62.77C23  C127D12  Dic33D12  Dic65D9  Dic9.D6  C12.S32  D6.3S32  Dic3.S32  C12.40S32  C12.58S32  C12⋊S312S3

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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