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G = D6⋊D6order 144 = 24·32

2nd semidirect product of D6 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D62D6, C122D6, D124S3, C42S32, C3⋊S32D4, C32(S3×D4), C323(C2×D4), (C3×D12)⋊7C2, D6⋊S33C2, (S3×C6)⋊2C22, (C3×C12)⋊2C22, C6.9(C22×S3), (C3×C6).9C23, C3⋊Dic33C22, (C2×S32)⋊2C2, (C4×C3⋊S3)⋊2C2, C2.11(C2×S32), (C2×C3⋊S3).15C22, SmallGroup(144,145)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D6⋊D6
C1C3C32C3×C6S3×C6C2×S32 — D6⋊D6
C32C3×C6 — D6⋊D6
C1C2C4

Generators and relations for D6⋊D6
 G = < a,b,c,d | a6=b2=c6=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd=a3b, dcd=c-1 >

Subgroups: 448 in 116 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×10], C6 [×2], C6 [×5], C2×C4, D4 [×4], C23 [×2], C32, Dic3 [×3], C12 [×2], C12, D6 [×4], D6 [×11], C2×C6 [×4], C2×D4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×3], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×4], C2×C3⋊S3, S3×D4 [×2], D6⋊S3 [×2], C3×D12 [×2], C4×C3⋊S3, C2×S32 [×2], D6⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], S32, S3×D4 [×2], C2×S32, D6⋊D6

Character table of D6⋊D6

 class 12A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G12A12B12C12D
 size 11666699224218224121212124444
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-11111-1111-111-11111    linear of order 2
ρ311-1-111-1-11111-11111-1-111111    linear of order 2
ρ411-1-1-1-11111111111-1-1-1-11111    linear of order 2
ρ511-111-1-1-1111-111111-11-1-1-1-1-1    linear of order 2
ρ611-11-1111111-1-1111-1-111-1-1-1-1    linear of order 2
ρ7111-11-111111-1-111111-1-1-1-1-1-1    linear of order 2
ρ8111-1-11-1-1111-11111-11-11-1-1-1-1    linear of order 2
ρ9222200002-1-1202-1-10-1-10-12-1-1    orthogonal lifted from S3
ρ1022-2200002-1-1-202-1-101-101-211    orthogonal lifted from D6
ρ112-200002-222200-2-2-200000000    orthogonal lifted from D4
ρ1222002200-12-120-12-1-100-12-1-1-1    orthogonal lifted from S3
ρ132200-2200-12-1-20-12-1100-1-2111    orthogonal lifted from D6
ρ142-20000-2222200-2-2-200000000    orthogonal lifted from D4
ρ1522002-200-12-1-20-12-1-1001-2111    orthogonal lifted from D6
ρ162200-2-200-12-120-12-110012-1-1-1    orthogonal lifted from D6
ρ1722-2-200002-1-1202-1-10110-12-1-1    orthogonal lifted from D6
ρ18222-200002-1-1-202-1-10-1101-211    orthogonal lifted from D6
ρ1944000000-2-21-40-2-21000022-1-1    orthogonal lifted from C2×S32
ρ204-40000004-2-200-42200000000    orthogonal lifted from S3×D4
ρ2144000000-2-2140-2-210000-2-211    orthogonal lifted from S32
ρ224-4000000-24-2002-4200000000    orthogonal lifted from S3×D4
ρ234-4000000-2-210022-10000003i-3i    complex faithful
ρ244-4000000-2-210022-1000000-3i3i    complex faithful

Permutation representations of D6⋊D6
On 24 points - transitive group 24T231
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)
(1 23 5 19 3 21)(2 22 6 24 4 20)(7 18 11 14 9 16)(8 17 12 13 10 15)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)

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

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

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

G:=TransitiveGroup(24,231);

D6⋊D6 is a maximal subgroup of
C4.S3≀C2  C3⋊S3.2D8  C249D6  C244D6  C246D6  D12⋊D6  D12.D6  D12.9D6  D12.10D6  C4.4S3≀C2  C32⋊D8⋊C2  C3⋊S3⋊D8  C3⋊S32SD16  S32⋊D4  C4⋊S3≀C2  D1223D6  D1224D6  S32×D4  D1212D6  D1215D6  D1216D6  C36⋊D6  C3⋊S3⋊D12  D6⋊S32  (S3×C6)⋊D6  C12⋊S32  C123S32
D6⋊D6 is a maximal quotient of
C249D6  C244D6  C246D6  C24.23D6  D12.2D6  D245S3  D12.4D6  C62.24C23  C62.55C23  D12⋊Dic3  D64Dic6  C62.70C23  C62.72C23  C62.82C23  C62.84C23  C62.85C23  C122D12  C12⋊Dic6  C62.91C23  D64D12  C36⋊D6  C12.86S32  D6⋊S32  (S3×C6)⋊D6  C12⋊S32  C123S32

Matrix representation of D6⋊D6 in GL4(𝔽5) generated by

0300
3100
0012
0020
,
0003
0031
1200
2000
,
0001
0012
2000
0200
,
0020
0043
3000
1200
G:=sub<GL(4,GF(5))| [0,3,0,0,3,1,0,0,0,0,1,2,0,0,2,0],[0,0,1,2,0,0,2,0,0,3,0,0,3,1,0,0],[0,0,2,0,0,0,0,2,0,1,0,0,1,2,0,0],[0,0,3,1,0,0,0,2,2,4,0,0,0,3,0,0] >;

D6⋊D6 in GAP, Magma, Sage, TeX

D_6\rtimes D_6
% in TeX

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

G:=SmallGroup(144,145);
// by ID

G=gap.SmallGroup(144,145);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,116,50,490,3461]);
// Polycyclic

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

Export

Character table of D6⋊D6 in TeX

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