direct product, p-group, abelian, monomial
Aliases: C2×C16, SmallGroup(32,16)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C16 |
| C1 — C2×C16 |
| C1 — C2×C16 |
Generators and relations for C2×C16
G = < a,b | a2=b16=1, ab=ba >
Smallest permutation representation of C2×C16
►Regular action on 32 points
Generators in S32
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 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)
G:=sub<Sym(32)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (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)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (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) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32)], [(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)]])
C2×C16 is a maximal subgroup of
C16⋊5C4 C22⋊C16 D4.C8 C2.D16 C2.Q32 D8.C4 C4⋊C16 C16⋊3C4 C16⋊4C4 C8.4Q8 M6(2) D4○C16 C4○D16 D5⋊C16 C3⋊S3⋊3C16 C4.3F9 D13⋊C16
C2×C16 is a maximal quotient of
C22⋊C16 C4⋊C16 M6(2) D5⋊C16 C3⋊S3⋊3C16 C4.3F9 D13⋊C16
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 |
| kernel | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C16 ►in GL2(𝔽17) generated by
| 16 | 0 |
| 0 | 16 |
| 8 | 0 |
| 0 | 12 |
G:=sub<GL(2,GF(17))| [16,0,0,16],[8,0,0,12] >;
C2×C16 in GAP, Magma, Sage, TeX
C_2\times C_{16} % in TeX
G:=Group("C2xC16"); // GroupNames label
G:=SmallGroup(32,16);
// by ID
G=gap.SmallGroup(32,16);
# by ID
G:=PCGroup([5,-2,2,-2,-2,-2,20,42,58]);
// Polycyclic
G:=Group<a,b|a^2=b^16=1,a*b=b*a>;
// generators/relations
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