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## G = C2×C16order 32 = 25

### Abelian group of type [2,16]

Aliases: C2×C16, SmallGroup(32,16)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16
 Lower central C1 — C2×C16
 Upper central C1 — C2×C16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — 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
C165C4  C22⋊C16  D4.C8  C2.D16  C2.Q32  D8.C4  C4⋊C16  C163C4  C164C4  C8.4Q8  M6(2)  D4○C16  C4○D16  D5⋊C16  C3⋊S33C16  C4.3F9  D13⋊C16
C2×C16 is a maximal quotient of
C22⋊C16  C4⋊C16  M6(2)  D5⋊C16  C3⋊S33C16  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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