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## G = D5×C30order 300 = 22·3·52

### Direct product of C30 and D5

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

 Derived series C1 — C5 — D5×C30
 Chief series C1 — C5 — C52 — C5×C15 — D5×C15 — D5×C30
 Lower central C5 — D5×C30
 Upper central C1 — C30

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

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 2A 2B 2C 3A 3B 5A 5B 5C 5D 5E ··· 5N 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 15A ··· 15H 15I ··· 15AB 30A ··· 30H 30I ··· 30AB 30AC ··· 30AR order 1 2 2 2 3 3 5 5 5 5 5 ··· 5 6 6 6 6 6 6 10 10 10 10 10 ··· 10 10 ··· 10 15 ··· 15 15 ··· 15 30 ··· 30 30 ··· 30 30 ··· 30 size 1 1 5 5 1 1 1 1 1 1 2 ··· 2 1 1 5 5 5 5 1 1 1 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 D5 D10 C3×D5 C5×D5 C6×D5 D5×C10 D5×C15 D5×C30 kernel D5×C30 D5×C15 C5×C30 D5×C10 C6×D5 C5×D5 C5×C10 C3×D5 C30 D10 D5 C10 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 2 1 2 4 4 2 8 4 8 16 8 2 2 4 8 4 8 16 16

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

 12 0 0 12
,
 19 30 20 30
,
 30 0 11 1
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

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