p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C42)⋊8C4, C4.9C42⋊8C2, (C2×D4).268D4, (C2×Q8).210D4, C42⋊6C4⋊15C2, C42⋊C2⋊20C4, C42.141(C2×C4), C23.125(C2×D4), C4.85(C42⋊C2), C23.33(C22⋊C4), C42⋊C2.4C22, (C2×C42).253C22, (C22×C4).669C23, C4.1(C22.D4), C23.C23.4C2, C2.15(C23.34D4), (C2×M4(2)).163C22, C22.1(C22.D4), M4(2).8C22.6C2, (C4×C4○D4).8C2, (C2×C4).234(C2×D4), (C22×C4).76(C2×C4), (C2×C4).744(C4○D4), (C2×C4).46(C22⋊C4), (C2×C4).534(C22×C4), C22.35(C2×C22⋊C4), (C2×C4○D4).261C22, SmallGroup(128,559)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C42)⋊C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, dad-1=ac2, bc=cb, dbd-1=bc2, dcd-1=abc >
Subgroups: 260 in 131 conjugacy classes, 50 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C4.D4, C4.10D4, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C4.9C42, C42⋊6C4, C23.C23, M4(2).8C22, C4×C4○D4, (C2×C42)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, (C2×C42)⋊C4
►On 16 points - transitive group 16T291
(1 10)(2 11)(3 12)(4 9)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 2 3 4)(5 13)(6 14)(7 15)(8 16)(9 10 11 12)
(1 16)(2 15 4 13)(3 14)(5 11)(6 12 8 10)(7 9)
G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,9)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2,3,4)(5,13)(6,14)(7,15)(8,16)(9,10,11,12), (1,16)(2,15,4,13)(3,14)(5,11)(6,12,8,10)(7,9)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2,3,4)(5,13)(6,14)(7,15)(8,16)(9,10,11,12), (1,16)(2,15,4,13)(3,14)(5,11)(6,12,8,10)(7,9) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,2,3,4),(5,13),(6,14),(7,15),(8,16),(9,10,11,12)], [(1,16),(2,15,4,13),(3,14),(5,11),(6,12,8,10),(7,9)]])
G:=TransitiveGroup(16,291);
►On 16 points - transitive group 16T314
(1 3)(2 4)(5 7)(6 8)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 8 7 6)(9 11)(10 12)(13 16 15 14)
(1 5 9 14)(2 6 12 13)(3 7 11 16)(4 8 10 15)
G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,8,7,6)(9,11)(10,12)(13,16,15,14), (1,5,9,14)(2,6,12,13)(3,7,11,16)(4,8,10,15)>;
G:=Group( (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,8,7,6)(9,11)(10,12)(13,16,15,14), (1,5,9,14)(2,6,12,13)(3,7,11,16)(4,8,10,15) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(5,8,7,6),(9,11),(10,12),(13,16,15,14)], [(1,5,9,14),(2,6,12,13),(3,7,11,16),(4,8,10,15)]])
G:=TransitiveGroup(16,314);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | (C2×C42)⋊C4 |
| kernel | (C2×C42)⋊C4 | C4.9C42 | C42⋊6C4 | C23.C23 | M4(2).8C22 | C4×C4○D4 | C2×C42 | C42⋊C2 | C2×D4 | C2×Q8 | C2×C4 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 3 | 1 | 8 | 4 |
Matrix representation of (C2×C42)⋊C4 ►in GL4(𝔽5) generated by
| 0 | 0 | 1 | 3 |
| 0 | 0 | 4 | 0 |
| 0 | 4 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 2 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 2 | 1 | 0 | 0 |
| 3 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,2,0,0,4,2,1,4,0,0,3,0,0,0],[0,0,0,4,0,0,3,4,2,3,0,0,1,0,0,0],[0,0,2,1,0,0,2,3,4,0,0,0,0,4,0,0],[0,0,2,3,0,0,1,3,0,3,0,0,4,0,0,0] >;
(C2×C42)⋊C4 in GAP, Magma, Sage, TeX
(C_2\times C_4^2)\rtimes C_4
% in TeX
G:=Group("(C2xC4^2):C4"); // GroupNames label
G:=SmallGroup(128,559);
// by ID
G=gap.SmallGroup(128,559);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2019,172,2028,1027]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^2,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=a*b*c>;
// generators/relations