p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8.1C16, C16.6Q8, C16.30D4, C42.9C8, M6(2).4C2, C8.22M4(2), C22.5M5(2), C4.8(C2×C16), (C4×C8).28C4, (C2×C8).13C8, C2.5(C4⋊C16), C4.26(C4⋊C8), C8.43(C4⋊C4), (C2×C16).11C4, (C4×C16).17C2, (C2×C16).102C22, (C2×C4).78(C2×C8), (C2×C8).242(C2×C4), SmallGroup(128,154)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.C16
G = < a,b | a8=1, b16=a4, bab-1=a-1 >
Smallest permutation representation of C8.C16
►On 32 points
Generators in S32
(1 5 9 13 17 21 25 29)(2 30 26 22 18 14 10 6)(3 7 11 15 19 23 27 31)(4 32 28 24 20 16 12 8)
(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)
G:=sub<Sym(32)| (1,5,9,13,17,21,25,29)(2,30,26,22,18,14,10,6)(3,7,11,15,19,23,27,31)(4,32,28,24,20,16,12,8), (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)>;
G:=Group( (1,5,9,13,17,21,25,29)(2,30,26,22,18,14,10,6)(3,7,11,15,19,23,27,31)(4,32,28,24,20,16,12,8), (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) );
G=PermutationGroup([[(1,5,9,13,17,21,25,29),(2,30,26,22,18,14,10,6),(3,7,11,15,19,23,27,31),(4,32,28,24,20,16,12,8)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | ··· | 4G | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 16A | ··· | 16H | 16I | ··· | 16T | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | Q8 | M4(2) | M5(2) | C8.C16 |
| kernel | C8.C16 | C4×C16 | M6(2) | C4×C8 | C2×C16 | C42 | C2×C8 | C8 | C16 | C16 | C8 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 16 |
Matrix representation of C8.C16 ►in GL2(𝔽17) generated by
| 15 | 0 |
| 0 | 8 |
| 0 | 12 |
| 1 | 0 |
G:=sub<GL(2,GF(17))| [15,0,0,8],[0,1,12,0] >;
C8.C16 in GAP, Magma, Sage, TeX
C_8.C_{16} % in TeX
G:=Group("C8.C16"); // GroupNames label
G:=SmallGroup(128,154);
// by ID
G=gap.SmallGroup(128,154);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,36,1430,352,80,102,124]);
// Polycyclic
G:=Group<a,b|a^8=1,b^16=a^4,b*a*b^-1=a^-1>;
// generators/relations
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