p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8≀C2, D8⋊2C8, C82⋊8C2, Q16⋊2C8, C8.35D8, C8.23SD16, C8.9M4(2), C42.391D4, C8.9(C2×C8), C4○D8.1C4, C8○D8.1C2, C8.C8⋊6C2, C22.1C4≀C2, (C2×C8).282D4, C2.10(D4⋊C8), C8.C4.1C4, C4.5(C22⋊C8), (C4×C8).418C22, C4.47(D4⋊C4), (C2×C8).177(C2×C4), (C2×C4).215(C22⋊C4), 2-Sylow(GL(2,9)), SmallGroup(128,67)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8≀C2
G = < a,b,c | a8=b2=c8=1, bab=a-1, ac=ca, cbc-1=a-1b >
Permutation representations of C8≀C2
►On 16 points - transitive group 16T289
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)
(1 3 5 7)(2 4 6 8)(9 14 11 16 13 10 15 12)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13), (1,3,5,7)(2,4,6,8)(9,14,11,16,13,10,15,12)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13), (1,3,5,7)(2,4,6,8)(9,14,11,16,13,10,15,12) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13)], [(1,3,5,7),(2,4,6,8),(9,14,11,16,13,10,15,12)]])
G:=TransitiveGroup(16,289);
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 8A | 8B | 8C | 8D | 8E | ··· | 8Z | 8AA | 8AB | 16A | 16B | 16C | 16D |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 16 | 16 | 16 | 16 |
| size | 1 | 1 | 2 | 8 | 1 | 1 | 2 | ··· | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 | C8≀C2 |
| kernel | C8≀C2 | C82 | C8.C8 | C8○D8 | C8.C4 | C4○D8 | D8 | Q16 | C42 | C2×C8 | C8 | C8 | C8 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 16 |
Matrix representation of C8≀C2 ►in GL2(𝔽17) generated by
| 15 | 0 |
| 0 | 8 |
| 0 | 8 |
| 15 | 0 |
| 13 | 0 |
| 0 | 8 |
G:=sub<GL(2,GF(17))| [15,0,0,8],[0,15,8,0],[13,0,0,8] >;
C8≀C2 in GAP, Magma, Sage, TeX
C_8\wr C_2
% in TeX
G:=Group("C8wrC2"); // GroupNames label
G:=SmallGroup(128,67);
// by ID
G=gap.SmallGroup(128,67);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,219,436,136,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations
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