Aliases: C62⋊1C4, C3⋊S3.5D4, C22⋊(C32⋊C4), C32⋊2(C22⋊C4), (C2×C3⋊S3)⋊3C4, (C2×C32⋊C4)⋊2C2, (C3×C6).7(C2×C4), C2.7(C2×C32⋊C4), (C22×C3⋊S3).3C2, (C2×C3⋊S3).10C22, SmallGroup(144,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊C4
G = < a,b,c | a6=b6=c4=1, ab=ba, cac-1=a-1b, cbc-1=a4b >
Subgroups: 342 in 66 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C32, D6, C2×C6, C22⋊C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, C2×C32⋊C4, C22×C3⋊S3, C62⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4
Character table of C62⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 0 | -2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 3 | 2 | -1 | -3 | 0 | 0 | orthogonal faithful |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -3 | 2 | -1 | 3 | 0 | 0 | orthogonal faithful |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -3 | 3 | orthogonal faithful |
| ρ14 | 4 | 4 | 4 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | -2 | 1 | 1 | orthogonal lifted from C32⋊C4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 3 | -3 | orthogonal faithful |
| ρ16 | 4 | 4 | -4 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | 1 | -2 | 2 | -1 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ17 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -2 | -2 | orthogonal lifted from C32⋊C4 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | -2 | 1 | -1 | 2 | 2 | orthogonal lifted from C2×C32⋊C4 |
►On 12 points - transitive group 12T82
Generators in S12
(1 2 3)(4 5 6)(7 8 9 10 11 12)
(1 4 2 5 3 6)(7 10)(8 11)(9 12)
(1 10)(2 8 3 12)(4 9 6 11)(5 7)
G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9,10,11,12), (1,4,2,5,3,6)(7,10)(8,11)(9,12), (1,10)(2,8,3,12)(4,9,6,11)(5,7)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9,10,11,12), (1,4,2,5,3,6)(7,10)(8,11)(9,12), (1,10)(2,8,3,12)(4,9,6,11)(5,7) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9,10,11,12)], [(1,4,2,5,3,6),(7,10),(8,11),(9,12)], [(1,10),(2,8,3,12),(4,9,6,11),(5,7)]])
G:=TransitiveGroup(12,82);
►On 24 points - transitive group 24T272
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 4)(2 5)(3 6)(7 11)(8 12)(9 10)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 19 9 13)(2 21 8 17)(3 23 7 15)(4 22 10 16)(5 24 12 14)(6 20 11 18)
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), (1,4)(2,5)(3,6)(7,11)(8,12)(9,10)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,19,9,13)(2,21,8,17)(3,23,7,15)(4,22,10,16)(5,24,12,14)(6,20,11,18)>;
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), (1,4)(2,5)(3,6)(7,11)(8,12)(9,10)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,19,9,13)(2,21,8,17)(3,23,7,15)(4,22,10,16)(5,24,12,14)(6,20,11,18) );
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)], [(1,4),(2,5),(3,6),(7,11),(8,12),(9,10),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,19,9,13),(2,21,8,17),(3,23,7,15),(4,22,10,16),(5,24,12,14),(6,20,11,18)]])
G:=TransitiveGroup(24,272);
►On 24 points - transitive group 24T273
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 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 22 11 20 9 24)(8 23 12 21 10 19)
(1 22 14 9)(2 8 13 23)(3 20 18 11)(4 12 17 19)(5 24 16 7)(6 10 15 21)
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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,22,11,20,9,24)(8,23,12,21,10,19), (1,22,14,9)(2,8,13,23)(3,20,18,11)(4,12,17,19)(5,24,16,7)(6,10,15,21)>;
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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,22,11,20,9,24)(8,23,12,21,10,19), (1,22,14,9)(2,8,13,23)(3,20,18,11)(4,12,17,19)(5,24,16,7)(6,10,15,21) );
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)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(7,22,11,20,9,24),(8,23,12,21,10,19)], [(1,22,14,9),(2,8,13,23),(3,20,18,11),(4,12,17,19),(5,24,16,7),(6,10,15,21)]])
G:=TransitiveGroup(24,273);
►On 24 points - transitive group 24T274
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 7 3 11 9 6)(2 8 4 12 10 5)(13 20 15 22 17 24)(14 21 16 23 18 19)
(1 13 2 19)(3 15 10 23)(4 21 9 17)(5 18 7 20)(6 24 8 14)(11 22 12 16)
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), (1,7,3,11,9,6)(2,8,4,12,10,5)(13,20,15,22,17,24)(14,21,16,23,18,19), (1,13,2,19)(3,15,10,23)(4,21,9,17)(5,18,7,20)(6,24,8,14)(11,22,12,16)>;
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), (1,7,3,11,9,6)(2,8,4,12,10,5)(13,20,15,22,17,24)(14,21,16,23,18,19), (1,13,2,19)(3,15,10,23)(4,21,9,17)(5,18,7,20)(6,24,8,14)(11,22,12,16) );
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)], [(1,7,3,11,9,6),(2,8,4,12,10,5),(13,20,15,22,17,24),(14,21,16,23,18,19)], [(1,13,2,19),(3,15,10,23),(4,21,9,17),(5,18,7,20),(6,24,8,14),(11,22,12,16)]])
G:=TransitiveGroup(24,274);
C62⋊C4 is a maximal subgroup of
C62.2D4 (C6×C12)⋊C4 (C2×C62)⋊C4 C62.9D4 D6≀C2 C62⋊Q8 (C6×C12)⋊5C4 D4×C32⋊C4 D6⋊(C32⋊C4) C62⋊11Dic3 C62⋊Dic3
C62⋊C4 is a maximal quotient of
(C6×C12)⋊C4 C62.6(C2×C4) C3⋊Dic3.D4 (C6×C12)⋊2C4 C3⋊S3.5D8 C32⋊6C4≀C2 C3⋊S3.5Q16 C32⋊7C4≀C2 (C2×C62)⋊C4 C62⋊3C8 (C2×C62).C4 C22⋊(He3⋊C4) D6⋊(C32⋊C4) C62⋊11Dic3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T82 | x12-6x11+20x10-45x9+72x8-84x7+67x6-30x5+9x3-4x+1 | -39·138·172 |
Matrix representation of C62⋊C4 ►in GL4(ℤ) generated by
| -1 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 |
G:=sub<GL(4,Integers())| [-1,1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,-1],[1,-1,0,0,1,0,0,0,0,0,1,-1,0,0,1,0],[0,0,1,-1,0,0,0,-1,1,0,0,0,0,1,0,0] >;
C62⋊C4 in GAP, Magma, Sage, TeX
C_6^2\rtimes C_4
% in TeX
G:=Group("C6^2:C4"); // GroupNames label
G:=SmallGroup(144,136);
// by ID
G=gap.SmallGroup(144,136);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,3364,256,4613,881]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;
// generators/relations
Export