direct product, metabelian, soluble, monomial
Aliases: C2×C62⋊C4, C62⋊2(C2×C4), (C2×C62)⋊5C4, C23⋊2(C32⋊C4), (C22×C3⋊S3)⋊8C4, (C2×C3⋊S3).69D4, C3⋊S3.12(C2×D4), C22⋊2(C2×C32⋊C4), C32⋊4(C2×C22⋊C4), C3⋊S3⋊4(C22⋊C4), (C3×C6)⋊2(C22⋊C4), (C23×C3⋊S3).5C2, (C22×C32⋊C4)⋊4C2, (C2×C32⋊C4)⋊3C22, (C2×C3⋊S3).39C23, (C3×C6).36(C22×C4), C2.13(C22×C32⋊C4), (C22×C3⋊S3).98C22, (C2×C3⋊S3)⋊18(C2×C4), SmallGroup(288,941)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C62⋊C4
G = < a,b,c,d | a2=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >
Subgroups: 1488 in 266 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2×C4, C23, C23, C32, D6, C2×C6, C22⋊C4, C22×C4, C24, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C22×S3, C22×C6, C2×C22⋊C4, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×C23, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C62⋊C4, C22×C32⋊C4, C23×C3⋊S3, C2×C62⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C22×C32⋊C4, C2×C62⋊C4
►On 24 points - transitive group 24T674
(1 6)(2 5)(3 12)(4 11)(7 10)(8 9)(13 16)(14 17)(15 18)(19 22)(20 23)(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)
(1 3 9 5 11 7)(2 4 10 6 12 8)(13 23 15 19 17 21)(14 24 16 20 18 22)
(1 13 7 21)(2 22 8 18)(3 23 11 17)(4 14 12 20)(5 19 9 15)(6 16 10 24)
G:=sub<Sym(24)| (1,6)(2,5)(3,12)(4,11)(7,10)(8,9)(13,16)(14,17)(15,18)(19,22)(20,23)(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), (1,3,9,5,11,7)(2,4,10,6,12,8)(13,23,15,19,17,21)(14,24,16,20,18,22), (1,13,7,21)(2,22,8,18)(3,23,11,17)(4,14,12,20)(5,19,9,15)(6,16,10,24)>;
G:=Group( (1,6)(2,5)(3,12)(4,11)(7,10)(8,9)(13,16)(14,17)(15,18)(19,22)(20,23)(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), (1,3,9,5,11,7)(2,4,10,6,12,8)(13,23,15,19,17,21)(14,24,16,20,18,22), (1,13,7,21)(2,22,8,18)(3,23,11,17)(4,14,12,20)(5,19,9,15)(6,16,10,24) );
G=PermutationGroup([[(1,6),(2,5),(3,12),(4,11),(7,10),(8,9),(13,16),(14,17),(15,18),(19,22),(20,23),(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)], [(1,3,9,5,11,7),(2,4,10,6,12,8),(13,23,15,19,17,21),(14,24,16,20,18,22)], [(1,13,7,21),(2,22,8,18),(3,23,11,17),(4,14,12,20),(5,19,9,15),(6,16,10,24)]])
G:=TransitiveGroup(24,674);
►On 24 points - transitive group 24T675
(1 7)(2 8)(3 9)(4 12)(5 10)(6 11)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 13 7 19)(2 15 9 23)(3 17 8 21)(4 14 11 24)(5 16 10 22)(6 18 12 20)
G:=sub<Sym(24)| (1,7)(2,8)(3,9)(4,12)(5,10)(6,11)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,13,7,19)(2,15,9,23)(3,17,8,21)(4,14,11,24)(5,16,10,22)(6,18,12,20)>;
G:=Group( (1,7)(2,8)(3,9)(4,12)(5,10)(6,11)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,13,7,19)(2,15,9,23)(3,17,8,21)(4,14,11,24)(5,16,10,22)(6,18,12,20) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,12),(5,10),(6,11),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,13,7,19),(2,15,9,23),(3,17,8,21),(4,14,11,24),(5,16,10,22),(6,18,12,20)]])
G:=TransitiveGroup(24,675);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 4 | 4 | 18 | ··· | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | C32⋊C4 | C2×C32⋊C4 | C62⋊C4 |
| kernel | C2×C62⋊C4 | C62⋊C4 | C22×C32⋊C4 | C23×C3⋊S3 | C22×C3⋊S3 | C2×C62 | C2×C3⋊S3 | C23 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 4 | 2 | 6 | 8 |
Matrix representation of C2×C62⋊C4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,12,0,0] >;
C2×C62⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_6^2\rtimes C_4
% in TeX
G:=Group("C2xC6^2:C4"); // GroupNames label
G:=SmallGroup(288,941);
// by ID
G=gap.SmallGroup(288,941);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations