Copied to
clipboard

G = C2×C30order 60 = 22·3·5

Abelian group of type [2,30]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C30, SmallGroup(60,13)

Series: Derived Chief Lower central Upper central

C1 — C2×C30
C1C5C15C30 — C2×C30
C1 — C2×C30
C1 — C2×C30

Generators and relations for C2×C30
 G = < a,b | a2=b30=1, ab=ba >


Smallest permutation representation of C2×C30
Regular action on 60 points
Generators in S60
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(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)

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

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

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

60 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D6A···6F10A···10L15A···15H30A···30X
order12223355556···610···1015···1530···30
size11111111111···11···11···11···1

60 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC2×C30C30C2×C10C2×C6C10C6C22C2
# reps1324612824

Matrix representation of C2×C30 in GL2(𝔽31) generated by

300
01
,
50
029
G:=sub<GL(2,GF(31))| [30,0,0,1],[5,0,0,29] >;

C2×C30 in GAP, Magma, Sage, TeX

C_2\times C_{30}
% in TeX

G:=Group("C2xC30");
// GroupNames label

G:=SmallGroup(60,13);
// by ID

G=gap.SmallGroup(60,13);
# by ID

G:=PCGroup([4,-2,-2,-3,-5]);
// Polycyclic

G:=Group<a,b|a^2=b^30=1,a*b=b*a>;
// generators/relations

׿
×
𝔽