p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)≀C2, C42⋊44D4, C24.162D4, C22⋊1C4≀C2, C4⋊D4⋊14C4, C4.116(C4×D4), C22⋊Q8⋊14C4, C42⋊6C4⋊25C2, (C22×C42)⋊11C2, C23.566(C2×D4), (C22×C4).552D4, C4.188(C4⋊D4), C24.4C4⋊27C2, C22.27C22≀C2, (C23×C4).675C22, C22.19C24.7C2, C23.123(C22⋊C4), (C22×C4).1372C23, C42⋊C2.25C22, (C2×C42).1063C22, C2.35(C23.23D4), (C2×M4(2)).185C22, C22.25(C22.D4), C4⋊C4⋊8(C2×C4), (C2×D4)⋊9(C2×C4), (C2×C4≀C2)⋊15C2, (C2×Q8)⋊9(C2×C4), C2.42(C2×C4≀C2), (C2×C4).1531(C2×D4), (C2×C4).569(C4○D4), (C22×C4).406(C2×C4), (C2×C4).390(C22×C4), (C2×C4○D4).21C22, (C2×C4).195(C22⋊C4), C22.271(C2×C22⋊C4), SmallGroup(128,628)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)≀C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 420 in 221 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4≀C2, C2×C42, C2×C42, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C2×M4(2), C23×C4, C23×C4, C2×C4○D4, C42⋊6C4, C24.4C4, C2×C4≀C2, C22×C42, C22.19C24, (C2×C4)≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4≀C2, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C2×C4≀C2, (C2×C4)≀C2
►On 16 points - transitive group 16T211
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 8 6)(2 3 7 5)(9 14 11 16)(10 15 12 13)
(1 10 7 14)(2 16 8 12)(3 9 6 13)(4 15 5 11)
(1 12)(2 14)(3 11)(4 13)(5 9)(6 15)(7 16)(8 10)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,8,6)(2,3,7,5)(9,14,11,16)(10,15,12,13), (1,10,7,14)(2,16,8,12)(3,9,6,13)(4,15,5,11), (1,12)(2,14)(3,11)(4,13)(5,9)(6,15)(7,16)(8,10)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,8,6)(2,3,7,5)(9,14,11,16)(10,15,12,13), (1,10,7,14)(2,16,8,12)(3,9,6,13)(4,15,5,11), (1,12)(2,14)(3,11)(4,13)(5,9)(6,15)(7,16)(8,10) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,8,6),(2,3,7,5),(9,14,11,16),(10,15,12,13)], [(1,10,7,14),(2,16,8,12),(3,9,6,13),(4,15,5,11)], [(1,12),(2,14),(3,11),(4,13),(5,9),(6,15),(7,16),(8,10)]])
G:=TransitiveGroup(16,211);
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4Z | 4AA | 4AB | 4AC | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | C4○D4 | C4≀C2 |
| kernel | (C2×C4)≀C2 | C42⋊6C4 | C24.4C4 | C2×C4≀C2 | C22×C42 | C22.19C24 | C4⋊D4 | C22⋊Q8 | C42 | C22×C4 | C24 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 3 | 1 | 4 | 16 |
Matrix representation of (C2×C4)≀C2 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0],[0,16,0,0,16,0,0,0,0,0,0,4,0,0,13,0] >;
(C2×C4)≀C2 in GAP, Magma, Sage, TeX
(C_2\times C_4)\wr C_2
% in TeX
G:=Group("(C2xC4)wrC2"); // GroupNames label
G:=SmallGroup(128,628);
// by ID
G=gap.SmallGroup(128,628);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,248,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations