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G = D5×C30order 300 = 22·3·52

Direct product of C30 and D5

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: D5×C30, C10⋊C30, C302C10, C5⋊(C2×C30), (C5×C10)⋊3C6, (C5×C30)⋊3C2, C524(C2×C6), C153(C2×C10), (C5×C15)⋊8C22, SmallGroup(300,44)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C30
C1C5C52C5×C15D5×C15 — D5×C30
C5 — D5×C30
C1C30

Generators and relations for D5×C30
 G = < a,b,c | a30=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
2C5
2C5
5C22
5C6
5C6
2C10
2C10
5C10
5C10
2C15
2C15
5C2×C6
5C2×C10
2C30
2C30
5C30
5C30
5C2×C30

Smallest permutation representation of D5×C30
On 60 points
Generators in S60
(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)
(1 25 19 13 7)(2 26 20 14 8)(3 27 21 15 9)(4 28 22 16 10)(5 29 23 17 11)(6 30 24 18 12)(31 37 43 49 55)(32 38 44 50 56)(33 39 45 51 57)(34 40 46 52 58)(35 41 47 53 59)(36 42 48 54 60)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 31)(28 32)(29 33)(30 34)

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

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

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

120 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D5E···5N6A6B6C6D6E6F10A10B10C10D10E···10N10O···10V15A···15H15I···15AB30A···30H30I···30AB30AC···30AR
order12223355555···56666661010101010···1010···1015···1515···1530···3030···3030···30
size11551111112···211555511112···25···51···12···21···12···25···5

120 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C5C6C6C10C10C15C30C30D5D10C3×D5C5×D5C6×D5D5×C10D5×C15D5×C30
kernelD5×C30D5×C15C5×C30D5×C10C6×D5C5×D5C5×C10C3×D5C30D10D5C10C30C15C10C6C5C3C2C1
# reps12124428481682248481616

Matrix representation of D5×C30 in GL2(𝔽31) generated by

120
012
,
1930
2030
,
300
111
G:=sub<GL(2,GF(31))| [12,0,0,12],[19,20,30,30],[30,11,0,1] >;

D5×C30 in GAP, Magma, Sage, TeX

D_5\times C_{30}
% in TeX

G:=Group("D5xC30");
// GroupNames label

G:=SmallGroup(300,44);
// by ID

G=gap.SmallGroup(300,44);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,6004]);
// Polycyclic

G:=Group<a,b,c|a^30=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C30 in TeX

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