direct product, p-group, abelian, monomial
Aliases: C22×C8, SmallGroup(32,36)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C22×C8 |
| C1 — C22×C8 |
| C1 — C22×C8 |
Generators and relations for C22×C8
G = < a,b,c | a2=b2=c8=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C8
►Regular action on 32 points
Generators in S32
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(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)]])
C22×C8 is a maximal subgroup of
C22.7C42 C22.4Q16 C4.C42 C22⋊C16 C8○2M4(2) (C22×C8)⋊C2 C23.24D4 C42.6C22 C23.25D4 C8⋊9D4 C8⋊8D4 C8⋊7D4 C8.18D4
C22×C8 is a maximal quotient of
C42.12C4 D4○C16
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | C8 |
| kernel | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 16 |
Matrix representation of C22×C8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 13 |
G:=sub<GL(3,GF(17))| [1,0,0,0,16,0,0,0,1],[1,0,0,0,16,0,0,0,16],[2,0,0,0,2,0,0,0,13] >;
C22×C8 in GAP, Magma, Sage, TeX
C_2^2\times C_8
% in TeX
G:=Group("C2^2xC8"); // GroupNames label
G:=SmallGroup(32,36);
// by ID
G=gap.SmallGroup(32,36);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,40,58]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^8=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export