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G = C2×C16order 32 = 25

Abelian group of type [2,16]

direct product, p-group, abelian, monomial

Aliases: C2×C16, SmallGroup(32,16)

Series: Derived Chief Lower central Upper central Jennings

C1 — C2×C16
C1C2C4C8C2×C8 — C2×C16
C1 — C2×C16
C1 — C2×C16
C1C2C2C2C2C4C4C8 — 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)])

32 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H16A···16P
order122244448···816···16
size111111111···11···1

32 irreducible representations

dim11111111
type+++
imageC1C2C2C4C4C8C8C16
kernelC2×C16C16C2×C8C8C2×C4C4C22C2
# reps121224416

Matrix representation of C2×C16 in GL2(𝔽17) generated by

160
016
,
80
012
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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