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G = D6.4D6order 144 = 24·32

4th non-split extension by D6 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D6.4D6, Dic3.4D6, C62.8C22, C3⋊D41S3, C22.2S32, (C2×C6).5D6, D6⋊S34C2, (S3×Dic3)⋊2C2, C322Q85C2, C326(C4○D4), C34(D42S3), (S3×C6).4C22, (C3×C6).12C23, C6.12(C22×S3), C3⋊Dic3.14C22, (C3×Dic3).5C22, C2.13(C2×S32), (C3×C3⋊D4)⋊2C2, (C2×C3⋊Dic3)⋊5C2, SmallGroup(144,148)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D6.4D6
C1C3C32C3×C6S3×C6S3×Dic3 — D6.4D6
C32C3×C6 — D6.4D6
C1C2C22

Generators and relations for D6.4D6
 G = < a,b,c,d | a6=b2=1, c6=d2=a3, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c5 >

Subgroups: 256 in 88 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×4], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×3], C4○D4, C3×S3 [×2], C3×C6, C3×C6, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×5], C3⋊D4 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×2], C62, D42S3 [×2], S3×Dic3 [×2], D6⋊S3, C322Q8, C3×C3⋊D4 [×2], C2×C3⋊Dic3, D6.4D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, D42S3 [×2], C2×S32, D6.4D6

Character table of D6.4D6

 class 12A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I12A12B
 size 11266224669918224444412121212
ρ1111111111111111111111111    trivial
ρ211-111111-1-1-1-1111-1-1-1-1111-1-1    linear of order 2
ρ3111-111111-1-1-1-11111111-11-11    linear of order 2
ρ411-1-11111-1111-111-1-1-1-11-111-1    linear of order 2
ρ5111-1-1111-1-11111111111-1-1-1-1    linear of order 2
ρ611-1-1-111111-1-1111-1-1-1-11-1-111    linear of order 2
ρ71111-1111-11-1-1-111111111-11-1    linear of order 2
ρ811-11-11111-111-111-1-1-1-111-1-11    linear of order 2
ρ9222022-1-1200002-1-1-12-1-10-10-1    orthogonal lifted from S3
ρ1022220-12-102000-12-12-1-1-1-10-10    orthogonal lifted from S3
ρ1122-2022-1-1-200002-111-21-10-101    orthogonal lifted from D6
ρ1222-220-12-10-2000-121-211-1-1010    orthogonal lifted from D6
ρ132220-22-1-1-200002-1-1-12-1-10101    orthogonal lifted from D6
ρ1422-20-22-1-1200002-111-21-1010-1    orthogonal lifted from D6
ρ1522-2-20-12-102000-121-211-110-10    orthogonal lifted from D6
ρ16222-20-12-10-2000-12-12-1-1-11010    orthogonal lifted from D6
ρ172-2000222002i-2i0-2-20000-20000    complex lifted from C4○D4
ρ182-200022200-2i2i0-2-20000-20000    complex lifted from C4○D4
ρ1944-400-2-2100000-2-2-122-110000    orthogonal lifted from C2×S32
ρ2044400-2-2100000-2-21-2-2110000    orthogonal lifted from S32
ρ214-4000-24-2000002-4000020000    symplectic lifted from D42S3, Schur index 2
ρ224-4000-2-210000022-3003-10000    symplectic faithful, Schur index 2
ρ234-40004-2-200000-42000020000    symplectic lifted from D42S3, Schur index 2
ρ244-4000-2-210000022300-3-10000    symplectic faithful, Schur index 2

Permutation representations of D6.4D6
On 24 points - transitive group 24T206
Generators in S24
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 24 22 20 18 16)
(1 6)(2 9)(3 8)(4 11)(5 10)(7 12)(13 18)(14 21)(15 20)(16 23)(17 22)(19 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 19 7 13)(2 24 8 18)(3 17 9 23)(4 22 10 16)(5 15 11 21)(6 20 12 14)

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

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

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

G:=TransitiveGroup(24,206);

D6.4D6 is a maximal subgroup of
Dic3≀C2  C62.12D4  C62.13D4  C62.15D4  D12.34D6  D1223D6  Dic6.24D6  S3×D42S3  D1212D6  C32⋊2+ 1+4  D18.4D6  C62.8D6  D6.4S32  D6⋊S3⋊S3  D6.6S32  C62.91D6  C62.96D6
D6.4D6 is a maximal quotient of
C62.8C23  Dic3.Dic6  C62.31C23  C62.32C23  C62.40C23  C62.48C23  D6.9D12  D62Dic6  C62.72C23  C62.85C23  C62.98C23  C62.99C23  C62.101C23  C62.57D4  C62.111C23  C62.112C23  C62.115C23  C627D4  C624Q8  D18.4D6  C62.9D6  D6.4S32  D6⋊S3⋊S3  D6.6S32  C62.91D6  C62.96D6

Matrix representation of D6.4D6 in GL4(𝔽5) generated by

1004
0120
0200
1000
,
0020
0004
3000
0400
,
0020
4000
3002
0400
,
2000
0040
0100
2003
G:=sub<GL(4,GF(5))| [1,0,0,1,0,1,2,0,0,2,0,0,4,0,0,0],[0,0,3,0,0,0,0,4,2,0,0,0,0,4,0,0],[0,4,3,0,0,0,0,4,2,0,0,0,0,0,2,0],[2,0,0,2,0,0,1,0,0,4,0,0,0,0,0,3] >;

D6.4D6 in GAP, Magma, Sage, TeX

D_6._4D_6
% in TeX

G:=Group("D6.4D6");
// GroupNames label

G:=SmallGroup(144,148);
// by ID

G=gap.SmallGroup(144,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^6=d^2=a^3,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^5>;
// generators/relations

Export

Character table of D6.4D6 in TeX

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