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G = C22×C16order 64 = 26

Abelian group of type [2,2,16]

direct product, p-group, abelian, monomial

Aliases: C22×C16, SmallGroup(64,183)

Series: Derived Chief Lower central Upper central Jennings

C1 — C22×C16
C1C2C4C8C2×C8C22×C8 — C22×C16
C1 — C22×C16
C1 — C22×C16
C1C2C2C2C2C4C4C8 — C22×C16

Generators and relations for C22×C16
 G = < a,b,c | a2=b2=c16=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C16
Regular action on 64 points
Generators in S64
(1 48)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)
(1 62)(2 63)(3 64)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 33)(32 34)
(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)| (1,48)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (1,62)(2,63)(3,64)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,33)(32,34), (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( (1,48)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (1,62)(2,63)(3,64)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,33)(32,34), (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([(1,48),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(1,62),(2,63),(3,64),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,33),(32,34)], [(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)])

C22×C16 is a maximal subgroup of
C22.7M5(2)  M5(2)⋊7C4  C8.7C42  C8.8C42  C8.9C42  C22⋊C32  C162M5(2)  (C2×D4).5C8  C23.24D8  C4⋊C4.7C8  C23.25D8  C169D4  C167D4  C16.19D4  C168D4
C22×C16 is a maximal quotient of
C42.13C8  D4○C32

64 conjugacy classes

class 1 2A···2G4A···4H8A···8P16A···16AF
order12···24···48···816···16
size11···11···11···11···1

64 irreducible representations

dim11111111
type+++
imageC1C2C2C4C4C8C8C16
kernelC22×C16C2×C16C22×C8C2×C8C22×C4C2×C4C23C22
# reps1616212432

Matrix representation of C22×C16 in GL3(𝔽17) generated by

1600
0160
0016
,
100
010
0016
,
500
090
001
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,1,0,0,0,16],[5,0,0,0,9,0,0,0,1] >;

C22×C16 in GAP, Magma, Sage, TeX

C_2^2\times C_{16}
% in TeX

G:=Group("C2^2xC16");
// GroupNames label

G:=SmallGroup(64,183);
// by ID

G=gap.SmallGroup(64,183);
# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,48,69,88]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^16=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C22×C16 in TeX

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