p-group, metabelian, nilpotent (class 4), monomial
Aliases: D8⋊4C8, Q16⋊4C8, C8.32D8, C8.29SD16, C8.3M4(2), C42.392D4, C8.3(C2×C8), C8⋊C8⋊1C2, C4○D8.3C4, C8○D8.2C2, C8.C8⋊7C2, C22.2C4≀C2, (C2×C8).180D4, C2.11(D4⋊C8), C8.C4.2C4, C4.6(C22⋊C8), (C4×C8).131C22, C4.48(D4⋊C4), (C2×C8).48(C2×C4), (C2×C4).216(C22⋊C4), SmallGroup(128,68)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.32D8
G = < a,b,c | a8=b8=1, c2=a, bab-1=a5, ac=ca, cbc-1=ab-1 >
Permutation representations of C8.32D8
►On 16 points - transitive group 16T260
Generators in S16
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)
(1 13 9 5)(2 16 6 4 10 8 14 12)(3 7 11 15)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16), (1,13,9,5)(2,16,6,4,10,8,14,12)(3,7,11,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16), (1,13,9,5)(2,16,6,4,10,8,14,12)(3,7,11,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16)], [(1,13,9,5),(2,16,6,4,10,8,14,12),(3,7,11,15)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,260);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 8A | 8B | 8C | 8D | 8E | ··· | 8N | 8O | 8P | 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 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 | C8.32D8 |
| kernel | C8.32D8 | C8⋊C8 | 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 | 4 |
Matrix representation of C8.32D8 ►in GL4(𝔽5) generated by
| 0 | 1 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 2 | 2 |
| 3 | 0 | 3 | 1 |
| 0 | 3 | 3 | 2 |
| 3 | 1 | 1 | 1 |
| 2 | 3 | 3 | 4 |
| 1 | 1 | 2 | 1 |
| 3 | 4 | 2 | 2 |
G:=sub<GL(4,GF(5))| [0,2,0,0,1,0,0,0,0,0,0,3,0,0,4,0],[0,0,3,0,0,0,0,3,2,2,3,3,1,2,1,2],[3,2,1,3,1,3,1,4,1,3,2,2,1,4,1,2] >;
C8.32D8 in GAP, Magma, Sage, TeX
C_8._{32}D_8 % in TeX
G:=Group("C8.32D8"); // GroupNames label
G:=SmallGroup(128,68);
// by ID
G=gap.SmallGroup(128,68);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,891,100,1018,136,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=a,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
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