Aliases: C42⋊1C6, C23.1A4, C42⋊C3⋊1C2, C42⋊2C2⋊C3, C22.3(C2×A4), SmallGroup(96,71)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊C6 |
Generators and relations for C42⋊C6
G = < a,b,c | a4=b4=c6=1, ab=ba, cac-1=b-1, cbc-1=a-1b >
Character table of C42⋊C6
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | |
| size | 1 | 3 | 4 | 16 | 16 | 6 | 6 | 12 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | -3 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ9 | 6 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex faithful |
| ρ10 | 6 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T184
Generators in S16
(1 9 4 15)(2 12 3 6)(5 13 14 16)(7 11 10 8)
(1 14 3 8)(2 11 4 5)(6 7 9 16)(10 15 13 12)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
G:=sub<Sym(16)| (1,9,4,15)(2,12,3,6)(5,13,14,16)(7,11,10,8), (1,14,3,8)(2,11,4,5)(6,7,9,16)(10,15,13,12), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16)>;
G:=Group( (1,9,4,15)(2,12,3,6)(5,13,14,16)(7,11,10,8), (1,14,3,8)(2,11,4,5)(6,7,9,16)(10,15,13,12), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16) );
G=PermutationGroup([[(1,9,4,15),(2,12,3,6),(5,13,14,16),(7,11,10,8)], [(1,14,3,8),(2,11,4,5),(6,7,9,16),(10,15,13,12)], [(2,3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)]])
G:=TransitiveGroup(16,184);
►On 24 points - transitive group 24T120
Generators in S24
(1 7 5 11)(2 18)(3 24 4 21)(6 15)(8 23)(9 16 10 13)(12 20)(14 19 17 22)
(1 17)(2 20 6 23)(3 10 4 9)(5 14)(7 22)(8 18 12 15)(11 19)(13 21 16 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)
G:=sub<Sym(24)| (1,7,5,11)(2,18)(3,24,4,21)(6,15)(8,23)(9,16,10,13)(12,20)(14,19,17,22), (1,17)(2,20,6,23)(3,10,4,9)(5,14)(7,22)(8,18,12,15)(11,19)(13,21,16,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,7,5,11)(2,18)(3,24,4,21)(6,15)(8,23)(9,16,10,13)(12,20)(14,19,17,22), (1,17)(2,20,6,23)(3,10,4,9)(5,14)(7,22)(8,18,12,15)(11,19)(13,21,16,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([[(1,7,5,11),(2,18),(3,24,4,21),(6,15),(8,23),(9,16,10,13),(12,20),(14,19,17,22)], [(1,17),(2,20,6,23),(3,10,4,9),(5,14),(7,22),(8,18,12,15),(11,19),(13,21,16,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])
G:=TransitiveGroup(24,120);
►On 24 points - transitive group 24T121
Generators in S24
(1 10 7 4)(2 23 8 15)(3 13)(5 20 11 18)(6 24)(9 21)(12 16)(14 19 22 17)
(1 14 7 22)(2 18)(3 12 9 6)(4 17 10 19)(5 23)(8 20)(11 15)(13 16 21 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)
G:=sub<Sym(24)| (1,10,7,4)(2,23,8,15)(3,13)(5,20,11,18)(6,24)(9,21)(12,16)(14,19,22,17), (1,14,7,22)(2,18)(3,12,9,6)(4,17,10,19)(5,23)(8,20)(11,15)(13,16,21,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,10,7,4)(2,23,8,15)(3,13)(5,20,11,18)(6,24)(9,21)(12,16)(14,19,22,17), (1,14,7,22)(2,18)(3,12,9,6)(4,17,10,19)(5,23)(8,20)(11,15)(13,16,21,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([[(1,10,7,4),(2,23,8,15),(3,13),(5,20,11,18),(6,24),(9,21),(12,16),(14,19,22,17)], [(1,14,7,22),(2,18),(3,12,9,6),(4,17,10,19),(5,23),(8,20),(11,15),(13,16,21,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])
G:=TransitiveGroup(24,121);
►On 24 points - transitive group 24T122
Generators in S24
(2 19 9 13)(3 20 10 14)(4 11)(5 16 12 22)(6 23 7 17)(15 21)
(1 18 8 24)(2 13 9 19)(3 10)(4 21 11 15)(5 16 12 22)(14 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)
G:=sub<Sym(24)| (2,19,9,13)(3,20,10,14)(4,11)(5,16,12,22)(6,23,7,17)(15,21), (1,18,8,24)(2,13,9,19)(3,10)(4,21,11,15)(5,16,12,22)(14,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (2,19,9,13)(3,20,10,14)(4,11)(5,16,12,22)(6,23,7,17)(15,21), (1,18,8,24)(2,13,9,19)(3,10)(4,21,11,15)(5,16,12,22)(14,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([[(2,19,9,13),(3,20,10,14),(4,11),(5,16,12,22),(6,23,7,17),(15,21)], [(1,18,8,24),(2,13,9,19),(3,10),(4,21,11,15),(5,16,12,22),(14,20)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])
G:=TransitiveGroup(24,122);
C42⋊C6 is a maximal subgroup of
C24.6A4 C42⋊C3⋊S3 (C4×C20)⋊C6
C42⋊C6 is a maximal quotient of C42⋊C12 C24.2A4 C23.19(C2×A4) C42⋊C18 C42⋊C3⋊S3 (C4×C20)⋊C6
Matrix representation of C42⋊C6 ►in GL6(𝔽13)
| 11 | 2 | 10 | 0 | 8 | 0 |
| 1 | 12 | 12 | 0 | 3 | 1 |
| 11 | 2 | 10 | 0 | 8 | 1 |
| 2 | 3 | 11 | 8 | 5 | 0 |
| 5 | 0 | 8 | 0 | 1 | 5 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 5 | 5 | 1 | 12 | 0 |
| 0 | 12 | 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 1 | 5 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 11 | 2 | 2 | 5 | 7 | 8 |
| 3 | 2 | 10 | 5 | 8 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(6,GF(13))| [11,1,11,2,5,0,2,12,2,3,0,12,10,12,10,11,8,0,0,0,0,8,0,0,8,3,8,5,1,0,0,1,1,0,5,0],[0,0,8,0,0,0,0,0,5,12,12,8,0,0,5,1,0,0,1,0,1,0,0,0,0,1,12,0,0,1,0,0,0,0,0,5],[0,1,0,11,3,0,0,0,1,2,2,0,1,0,0,2,10,0,0,0,0,5,5,12,0,0,0,7,8,0,0,0,0,8,0,0] >;
C42⋊C6 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_6
% in TeX
G:=Group("C4^2:C6"); // GroupNames label
G:=SmallGroup(96,71);
// by ID
G=gap.SmallGroup(96,71);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-2,2,542,116,230,1443,801,69,730,1307]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^6=1,a*b=b*a,c*a*c^-1=b^-1,c*b*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C42⋊C6 in TeX
Character table of C42⋊C6 in TeX