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G = C2×C30order 60 = 22·3·5

Abelian group of type [2,30]

Aliases: C2×C30, SmallGroup(60,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30
 Chief series C1 — C5 — C15 — C30 — C2×C30
 Lower central C1 — C2×C30
 Upper central 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 56)(2 57)(3 58)(4 59)(5 60)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)
(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)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55), (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( (1,56)(2,57)(3,58)(4,59)(5,60)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55), (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([(1,56),(2,57),(3,58),(4,59),(5,60),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55)], [(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)])

C2×C30 is a maximal subgroup of   C157D4

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 6A ··· 6F 10A ··· 10L 15A ··· 15H 30A ··· 30X order 1 2 2 2 3 3 5 5 5 5 6 ··· 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C5 C6 C10 C15 C30 kernel C2×C30 C30 C2×C10 C2×C6 C10 C6 C22 C2 # reps 1 3 2 4 6 12 8 24

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

 30 0 0 1
,
 5 0 0 29
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

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