p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C16.1C8, C8.31SD16, C42.14Q8, C8.13M4(2), C4.4(C4⋊C8), C8.20(C2×C8), (C2×C16).2C4, (C2×C8).181D4, C16⋊5C4.4C2, C2.4(C8⋊2C8), C8.C8.3C2, C4.14(C4.Q8), (C4×C8).136C22, C22.3(C8.C4), (C2×C8).224(C2×C4), (C2×C4).100(C4⋊C4), SmallGroup(128,101)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16.C8
G = < a,b | a16=1, b8=a8, bab-1=a3 >
Smallest permutation representation of C16.C8
►On 32 points
Generators in S32
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(1 31 7 17 13 19 3 21 9 23 15 25 5 27 11 29)(2 26 16 20 14 30 12 24 10 18 8 28 6 22 4 32)
G:=sub<Sym(32)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,31,7,17,13,19,3,21,9,23,15,25,5,27,11,29)(2,26,16,20,14,30,12,24,10,18,8,28,6,22,4,32)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,31,7,17,13,19,3,21,9,23,15,25,5,27,11,29)(2,26,16,20,14,30,12,24,10,18,8,28,6,22,4,32) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,31,7,17,13,19,3,21,9,23,15,25,5,27,11,29),(2,26,16,20,14,30,12,24,10,18,8,28,6,22,4,32)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 8A | ··· | 8H | 16A | ··· | 16H | 16I | ··· | 16P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | C8 | Q8 | D4 | M4(2) | SD16 | C8.C4 | C16.C8 |
| kernel | C16.C8 | C16⋊5C4 | C8.C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C8 | C22 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of C16.C8 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,9,0,0,13,0,0,0,0,0,0,1,0,0,9,0],[0,0,9,0,0,0,0,8,1,0,0,0,0,1,0,0] >;
C16.C8 in GAP, Magma, Sage, TeX
C_{16}.C_8 % in TeX
G:=Group("C16.C8"); // GroupNames label
G:=SmallGroup(128,101);
// by ID
G=gap.SmallGroup(128,101);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,36,422,100,1018,136,2804,172,124]);
// Polycyclic
G:=Group<a,b|a^16=1,b^8=a^8,b*a*b^-1=a^3>;
// generators/relations
Export