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G = C4×C16order 64 = 26

Abelian group of type [4,16]

direct product, p-group, abelian, monomial

Aliases: C4×C16, SmallGroup(64,26)

Series: Derived Chief Lower central Upper central Jennings

C1 — C4×C16
C1C2C4C2×C4C2×C8C4×C8 — C4×C16
C1 — C4×C16
C1 — C4×C16
C1C2C2C2C2C4C4C2×C8 — C4×C16

Generators and relations for C4×C16
 G = < a,b | a4=b16=1, ab=ba >


Smallest permutation representation of C4×C16
Regular action on 64 points
Generators in S64
(1 50 25 33)(2 51 26 34)(3 52 27 35)(4 53 28 36)(5 54 29 37)(6 55 30 38)(7 56 31 39)(8 57 32 40)(9 58 17 41)(10 59 18 42)(11 60 19 43)(12 61 20 44)(13 62 21 45)(14 63 22 46)(15 64 23 47)(16 49 24 48)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)

G:=sub<Sym(64)| (1,50,25,33)(2,51,26,34)(3,52,27,35)(4,53,28,36)(5,54,29,37)(6,55,30,38)(7,56,31,39)(8,57,32,40)(9,58,17,41)(10,59,18,42)(11,60,19,43)(12,61,20,44)(13,62,21,45)(14,63,22,46)(15,64,23,47)(16,49,24,48), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)>;

G:=Group( (1,50,25,33)(2,51,26,34)(3,52,27,35)(4,53,28,36)(5,54,29,37)(6,55,30,38)(7,56,31,39)(8,57,32,40)(9,58,17,41)(10,59,18,42)(11,60,19,43)(12,61,20,44)(13,62,21,45)(14,63,22,46)(15,64,23,47)(16,49,24,48), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64) );

G=PermutationGroup([(1,50,25,33),(2,51,26,34),(3,52,27,35),(4,53,28,36),(5,54,29,37),(6,55,30,38),(7,56,31,39),(8,57,32,40),(9,58,17,41),(10,59,18,42),(11,60,19,43),(12,61,20,44),(13,62,21,45),(14,63,22,46),(15,64,23,47),(16,49,24,48)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)])

C4×C16 is a maximal subgroup of
C165C8  C8⋊C16  D4⋊C16  C4.16D16  Q161C8  Q8⋊C16  C163C8  C164C8  C16.3C8  C325C4  C4⋊C32  C8.C16  C162M5(2)  C42.13C8  C8.12M4(2)  C166D4  C16○D8  C8○D16  C164Q8  C4.4D16  C4.SD32  C8.22SD16  C4⋊D16  C4⋊Q32  C165D4  C8.21D8  C162Q8  C16.5Q8  C163Q8  Dic5⋊C16
C4×C16 is a maximal quotient of
C8⋊C16  C22.7M5(2)  C325C4  Dic5⋊C16

64 conjugacy classes

class 1 2A2B2C4A···4L8A···8P16A···16AF
order12224···48···816···16
size11111···11···11···1

64 irreducible representations

dim111111111
type+++
imageC1C2C2C4C4C4C8C8C16
kernelC4×C16C4×C8C2×C16C16C42C2×C8C8C2×C4C4
# reps1128228832

Matrix representation of C4×C16 in GL2(𝔽17) generated by

130
01
,
110
010
G:=sub<GL(2,GF(17))| [13,0,0,1],[11,0,0,10] >;

C4×C16 in GAP, Magma, Sage, TeX

C_4\times C_{16}
% in TeX

G:=Group("C4xC16");
// GroupNames label

G:=SmallGroup(64,26);
// by ID

G=gap.SmallGroup(64,26);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,24,55,86,88]);
// Polycyclic

G:=Group<a,b|a^4=b^16=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C16 in TeX

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