p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C16.3C8, C8.37D8, C8.12Q16, C42.47Q8, C8.18M4(2), C4.7(C4⋊C8), C8.16(C2×C8), (C2×C16).13C4, (C4×C16).11C2, (C2×C8).283D4, C2.5(C8⋊1C8), C8.C8.4C2, C4.21(C2.D8), (C4×C8).419C22, C22.4(C8.C4), (C2×C8).225(C2×C4), (C2×C4).103(C4⋊C4), SmallGroup(128,105)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16.3C8
G = < a,b | a16=1, b8=a8, bab-1=a7 >
Smallest permutation representation of C16.3C8
►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 30 11 20 5 26 15 32 9 22 3 28 13 18 7 24)(2 21 12 27 6 17 16 23 10 29 4 19 14 25 8 31)
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), (1,30,11,20,5,26,15,32,9,22,3,28,13,18,7,24)(2,21,12,27,6,17,16,23,10,29,4,19,14,25,8,31)>;
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), (1,30,11,20,5,26,15,32,9,22,3,28,13,18,7,24)(2,21,12,27,6,17,16,23,10,29,4,19,14,25,8,31) );
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)], [(1,30,11,20,5,26,15,32,9,22,3,28,13,18,7,24),(2,21,12,27,6,17,16,23,10,29,4,19,14,25,8,31)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | ··· | 4G | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C4 | C8 | Q8 | D4 | M4(2) | D8 | Q16 | C8.C4 | C16.3C8 |
| kernel | C16.3C8 | C4×C16 | C8.C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C8 | C8 | C22 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 16 |
Matrix representation of C16.3C8 ►in GL2(𝔽17) generated by
| 11 | 6 |
| 0 | 3 |
| 14 | 10 |
| 4 | 3 |
G:=sub<GL(2,GF(17))| [11,0,6,3],[14,4,10,3] >;
C16.3C8 in GAP, Magma, Sage, TeX
C_{16}._3C_8 % in TeX
G:=Group("C16.3C8"); // GroupNames label
G:=SmallGroup(128,105);
// by ID
G=gap.SmallGroup(128,105);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,148,422,436,136,2804,172,124]);
// Polycyclic
G:=Group<a,b|a^16=1,b^8=a^8,b*a*b^-1=a^7>;
// generators/relations
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