non-abelian, soluble, monomial, rational
Aliases: C62⋊5D6, C3⋊S3⋊S4, C3⋊S4⋊S3, (C3×A4)⋊D6, C32⋊(C2×S4), C3.2(S3×S4), C62⋊S3⋊C2, C62⋊C6⋊C2, C32⋊S4⋊C2, C32⋊A4⋊C22, C22⋊(C32⋊D6), (C2×C6).2S32, (C22×C3⋊S3)⋊4S3, SmallGroup(432,523)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊A4 — C62⋊5D6 |
Generators and relations for C62⋊5D6
G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a2b-1, dad=a-1b-1, cbc-1=a3b-1, bd=db, dcd=c-1 >
Subgroups: 1289 in 134 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, S4, C2×A4, C22×S3, He3, C3×Dic3, S32, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×S4, C32⋊C6, He3⋊C2, C6.D6, C3⋊D12, C3×C3⋊D4, C3×S4, C3⋊S4, S3×A4, C2×S32, C22×C3⋊S3, C32⋊D6, C32⋊A4, Dic3⋊D6, S3×S4, C62⋊S3, C32⋊S4, C62⋊C6, C62⋊5D6
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, C32⋊D6, S3×S4, C62⋊5D6
Character table of C62⋊5D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | |
| size | 1 | 3 | 9 | 18 | 18 | 27 | 2 | 6 | 24 | 48 | 18 | 18 | 6 | 6 | 12 | 36 | 36 | 72 | 36 | 36 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 0 | -2 | 0 | 2 | -1 | 2 | -1 | -2 | 0 | 2 | -1 | -1 | 0 | 1 | 0 | 1 | 0 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | -1 | 2 | -1 | 2 | 0 | 2 | -1 | -1 | 0 | -1 | 0 | -1 | 0 | orthogonal lifted from S3 |
| ρ9 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ10 | 3 | -1 | -3 | 1 | -1 | 1 | 3 | 3 | 0 | 0 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 0 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ11 | 3 | -1 | 3 | 1 | 1 | -1 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ12 | 3 | -1 | -3 | -1 | 1 | 1 | 3 | 3 | 0 | 0 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 0 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ14 | 6 | -2 | 0 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | 1 | 4 | -2 | -1 | 0 | 0 | 0 | 1 | orthogonal faithful |
| ρ15 | 6 | -2 | 0 | 0 | 2 | 0 | 6 | -3 | 0 | 0 | -2 | 0 | -2 | 1 | 1 | 0 | -1 | 0 | 1 | 0 | orthogonal lifted from S3×S4 |
| ρ16 | 6 | -2 | 0 | 0 | -2 | 0 | 6 | -3 | 0 | 0 | 2 | 0 | -2 | 1 | 1 | 0 | 1 | 0 | -1 | 0 | orthogonal lifted from S3×S4 |
| ρ17 | 6 | 6 | 0 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | -3 | 0 | 0 | -1 | 0 | 0 | 0 | -1 | orthogonal lifted from C32⋊D6 |
| ρ18 | 6 | -2 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | 1 | 4 | -2 | 1 | 0 | 0 | 0 | -1 | orthogonal faithful |
| ρ19 | 6 | 6 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | -3 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | orthogonal lifted from C32⋊D6 |
| ρ20 | 12 | -4 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 18 points - transitive group 18T152
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 11 8 12 9 10)(13 17 15)(14 18 16)
(1 16 10)(2 13 8)(3 18 12 5 14 11)(4 15 7 6 17 9)
(1 8)(2 10)(3 7)(4 12)(5 9)(6 11)(13 16)(14 17)(15 18)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,11,8,12,9,10)(13,17,15)(14,18,16), (1,16,10)(2,13,8)(3,18,12,5,14,11)(4,15,7,6,17,9), (1,8)(2,10)(3,7)(4,12)(5,9)(6,11)(13,16)(14,17)(15,18)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,11,8,12,9,10)(13,17,15)(14,18,16), (1,16,10)(2,13,8)(3,18,12,5,14,11)(4,15,7,6,17,9), (1,8)(2,10)(3,7)(4,12)(5,9)(6,11)(13,16)(14,17)(15,18) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,11,8,12,9,10),(13,17,15),(14,18,16)], [(1,16,10),(2,13,8),(3,18,12,5,14,11),(4,15,7,6,17,9)], [(1,8),(2,10),(3,7),(4,12),(5,9),(6,11),(13,16),(14,17),(15,18)]])
G:=TransitiveGroup(18,152);
►On 18 points - transitive group 18T153
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 13 11 2 16 8)(3 15 10 6 14 9)(4 18 7 5 17 12)
(1 11)(2 8)(3 10)(4 7)(5 12)(6 9)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,13,11,2,16,8)(3,15,10,6,14,9)(4,18,7,5,17,12), (1,11)(2,8)(3,10)(4,7)(5,12)(6,9)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,13,11,2,16,8)(3,15,10,6,14,9)(4,18,7,5,17,12), (1,11)(2,8)(3,10)(4,7)(5,12)(6,9) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,13,11,2,16,8),(3,15,10,6,14,9),(4,18,7,5,17,12)], [(1,11),(2,8),(3,10),(4,7),(5,12),(6,9)]])
G:=TransitiveGroup(18,153);
►On 18 points - transitive group 18T154
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 17 9)(2 14 12)(3 13 8 5 15 7)(4 16 11 6 18 10)
(1 9)(2 12)(3 8)(4 11)(5 7)(6 10)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,17,9)(2,14,12)(3,13,8,5,15,7)(4,16,11,6,18,10), (1,9)(2,12)(3,8)(4,11)(5,7)(6,10)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,17,9)(2,14,12)(3,13,8,5,15,7)(4,16,11,6,18,10), (1,9)(2,12)(3,8)(4,11)(5,7)(6,10) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,17,9),(2,14,12),(3,13,8,5,15,7),(4,16,11,6,18,10)], [(1,9),(2,12),(3,8),(4,11),(5,7),(6,10)]])
G:=TransitiveGroup(18,154);
►On 18 points - transitive group 18T155
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 10 8 11 9 12)(13 17 15)(14 18 16)
(1 15 11 2 18 7)(3 17 10 6 16 8)(4 14 9 5 13 12)
(1 7)(2 11)(3 9)(4 10)(5 8)(6 12)(13 16)(14 17)(15 18)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,10,8,11,9,12)(13,17,15)(14,18,16), (1,15,11,2,18,7)(3,17,10,6,16,8)(4,14,9,5,13,12), (1,7)(2,11)(3,9)(4,10)(5,8)(6,12)(13,16)(14,17)(15,18)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,10,8,11,9,12)(13,17,15)(14,18,16), (1,15,11,2,18,7)(3,17,10,6,16,8)(4,14,9,5,13,12), (1,7)(2,11)(3,9)(4,10)(5,8)(6,12)(13,16)(14,17)(15,18) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,10,8,11,9,12),(13,17,15),(14,18,16)], [(1,15,11,2,18,7),(3,17,10,6,16,8),(4,14,9,5,13,12)], [(1,7),(2,11),(3,9),(4,10),(5,8),(6,12),(13,16),(14,17),(15,18)]])
G:=TransitiveGroup(18,155);
Matrix representation of C62⋊5D6 ►in GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0],[0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;
C62⋊5D6 in GAP, Magma, Sage, TeX
C_6^2\rtimes_5D_6
% in TeX
G:=Group("C6^2:5D6"); // GroupNames label
G:=SmallGroup(432,523);
// by ID
G=gap.SmallGroup(432,523);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,93,675,353,2524,1271,4548,2287,2659,3989]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^2*b^-1,d*a*d=a^-1*b^-1,c*b*c^-1=a^3*b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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