direct product, metabelian, soluble, monomial
Aliases: C2×C42⋊C6, C24.4A4, C42⋊2(C2×C6), (C2×C42)⋊1C6, C42⋊C3⋊3C22, C23.3(C2×A4), C42⋊2C2⋊2C6, C22.3(C22×A4), (C2×C42⋊2C2)⋊C3, (C2×C42⋊C3)⋊1C2, SmallGroup(192,1001)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C2×C42⋊C6 |
Generators and relations for C2×C42⋊C6
G = < a,b,c,d | a2=b4=c4=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=c-1, dcd-1=b-1c >
Subgroups: 282 in 66 conjugacy classes, 17 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C2×C4, C23, C23, C23, A4, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C24, C2×A4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C42⋊2C2, C42⋊C3, C22×A4, C2×C42⋊2C2, C2×C42⋊C3, C42⋊C6, C2×C42⋊C6
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C2×A4, C22×A4, C42⋊C6, C2×C42⋊C6
Character table of C2×C42⋊C6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 16 | 16 | 6 | 6 | 6 | 6 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | |
| 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 | 1 | -1 | -1 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | linear of order 6 |
| ρ6 | 1 | -1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | 1 | -1 | 1 | -1 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | -1 | -1 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | 1 | -1 | 1 | -1 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ11 | 1 | -1 | -1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | -1 | -1 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | linear of order 6 |
| ρ13 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ14 | 3 | -3 | -3 | 3 | 3 | -3 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ16 | 3 | -3 | -3 | 3 | -3 | 3 | 0 | 0 | 1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ17 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
| ρ18 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
| ρ20 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T290
Generators in S24
(1 12)(2 10)(3 11)(4 8)(5 9)(6 7)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 15)(2 20 10 23)(3 9 11 5)(4 16 8 13)(6 22)(7 19)(12 18)(14 24 17 21)
(1 22 12 19)(2 4 10 8)(3 14)(5 21)(6 18 7 15)(9 24)(11 17)(13 20 16 23)
(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)| (1,12)(2,10)(3,11)(4,8)(5,9)(6,7)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,15)(2,20,10,23)(3,9,11,5)(4,16,8,13)(6,22)(7,19)(12,18)(14,24,17,21), (1,22,12,19)(2,4,10,8)(3,14)(5,21)(6,18,7,15)(9,24)(11,17)(13,20,16,23), (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( (1,12)(2,10)(3,11)(4,8)(5,9)(6,7)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,15)(2,20,10,23)(3,9,11,5)(4,16,8,13)(6,22)(7,19)(12,18)(14,24,17,21), (1,22,12,19)(2,4,10,8)(3,14)(5,21)(6,18,7,15)(9,24)(11,17)(13,20,16,23), (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([[(1,12),(2,10),(3,11),(4,8),(5,9),(6,7),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,15),(2,20,10,23),(3,9,11,5),(4,16,8,13),(6,22),(7,19),(12,18),(14,24,17,21)], [(1,22,12,19),(2,4,10,8),(3,14),(5,21),(6,18,7,15),(9,24),(11,17),(13,20,16,23)], [(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(24,290);
►On 24 points - transitive group 24T300
Generators in S24
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 14 7 22)(2 20)(3 6 9 12)(4 17 10 19)(5 15)(8 18)(11 23)(13 24 21 16)
(1 19)(2 11 8 5)(3 24 9 16)(4 14)(6 21 12 13)(7 17)(10 22)(15 20 23 18)
(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)(2,8)(3,9)(4,10)(5,11)(6,12)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,14,7,22)(2,20)(3,6,9,12)(4,17,10,19)(5,15)(8,18)(11,23)(13,24,21,16), (1,19)(2,11,8,5)(3,24,9,16)(4,14)(6,21,12,13)(7,17)(10,22)(15,20,23,18), (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)(2,8)(3,9)(4,10)(5,11)(6,12)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,14,7,22)(2,20)(3,6,9,12)(4,17,10,19)(5,15)(8,18)(11,23)(13,24,21,16), (1,19)(2,11,8,5)(3,24,9,16)(4,14)(6,21,12,13)(7,17)(10,22)(15,20,23,18), (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),(2,8),(3,9),(4,10),(5,11),(6,12),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,14,7,22),(2,20),(3,6,9,12),(4,17,10,19),(5,15),(8,18),(11,23),(13,24,21,16)], [(1,19),(2,11,8,5),(3,24,9,16),(4,14),(6,21,12,13),(7,17),(10,22),(15,20,23,18)], [(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(24,300);
►On 24 points - transitive group 24T306
Generators in S24
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 20 8 18)(2 13 9 21)(4 15 11 23)(5 16 12 24)(6 7)(17 19)
(1 20 8 18)(3 22 10 14)(4 23 11 15)(5 12)(6 17 7 19)(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)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,20,8,18)(2,13,9,21)(4,15,11,23)(5,16,12,24)(6,7)(17,19), (1,20,8,18)(3,22,10,14)(4,23,11,15)(5,12)(6,17,7,19)(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( (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,20,8,18)(2,13,9,21)(4,15,11,23)(5,16,12,24)(6,7)(17,19), (1,20,8,18)(3,22,10,14)(4,23,11,15)(5,12)(6,17,7,19)(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([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,20,8,18),(2,13,9,21),(4,15,11,23),(5,16,12,24),(6,7),(17,19)], [(1,20,8,18),(3,22,10,14),(4,23,11,15),(5,12),(6,17,7,19),(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(24,306);
Matrix representation of C2×C42⋊C6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 12 | 8 | 12 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 | 1 | 0 |
| 4 | 5 | 5 | 0 | 0 | 5 |
| 7 | 7 | 12 | 0 | 0 | 11 |
| 5 | 0 | 0 | 11 | 0 | 0 |
| 0 | 5 | 0 | 0 | 11 | 0 |
| 11 | 11 | 10 | 0 | 0 | 8 |
| 12 | 0 | 0 | 8 | 0 | 0 |
| 3 | 2 | 3 | 6 | 1 | 6 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,1,0,12,0,5,0,0,0,8,0,0,1,0,0,12,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[8,0,0,0,0,4,0,12,0,0,8,5,0,0,8,0,0,5,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,5],[7,5,0,11,12,3,7,0,5,11,0,2,12,0,0,10,0,3,0,11,0,0,8,6,0,0,11,0,0,1,11,0,0,8,0,6] >;
C2×C42⋊C6 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes C_6
% in TeX
G:=Group("C2xC4^2:C6"); // GroupNames label
G:=SmallGroup(192,1001);
// by ID
G=gap.SmallGroup(192,1001);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,1683,185,360,4204,1173,102,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=c^-1,d*c*d^-1=b^-1*c>;
// generators/relations
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