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G = C2xC30order 60 = 22·3·5

Abelian group of type [2,30]

direct product, abelian, monomial, 2-elementary

Aliases: C2xC30, SmallGroup(60,13)

Series: Derived Chief Lower central Upper central

C1 — C2xC30
C1C5C15C30 — C2xC30
C1 — C2xC30
C1 — C2xC30

Generators and relations for C2xC30
 G = < a,b | a2=b30=1, ab=ba >

Subgroups: 20, all normal (8 characteristic)
Quotients: C1, C2, C3, C22, C5, C6, C10, C2xC6, C15, C2xC10, C30, C2xC30

Smallest permutation representation of C2xC30
Regular action on 60 points
Generators in S60
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)
(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)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50), (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( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50), (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([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50)], [(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)]])

C2xC30 is a maximal subgroup of   C15:7D4

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
kernelC2xC30C30C2xC10C2xC6C10C6C22C2
# reps1324612824

Matrix representation of C2xC30 in GL2(F31) generated by

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

C2xC30 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

Export

Subgroup lattice of C2xC30 in TeX

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