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G = D30⋊S3  order 360 = 23·32·5

3rd semidirect product of D30 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D30⋊3S3, C15⋊3D12, C30.19D6, Dic15⋊4S3, C10.10S32, (C3×C15)⋊13D4, (C6×D15)⋊7C2, C6.25(S3×D5), C5⋊3(C3⋊D12), C3⋊3(C5⋊D12), C3⋊1(C15⋊D4), C15⋊5(C3⋊D4), (C3×C6).10D10, C32⋊4(C5⋊D4), C2.3(D15⋊S3), (C3×Dic15)⋊10C2, (C3×C30).24C22, (C2×C3⋊S3)⋊3D5, (C10×C3⋊S3)⋊3C2, SmallGroup(360,86)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C30 — D30⋊S3
Chief series
C1 — C5 — C15 — C3×C15 — C3×C30 — C6×D15 — D30⋊S3
Lower central
C3×C15 — C3×C30 — D30⋊S3
Upper central
C1 — C2

Generators and relations for D30⋊S3
 G = < a,b,c,d | a30=b2=c3=d2=1, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a25b, dcd=c-1 >

Subgroups: 444 in 74 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3⋊S3, C3×C6, Dic5, D10, C2×C10, D12, C3⋊D4, C5×S3, C3×D5, D15, C30, C30, C3×Dic3, S3×C6, C2×C3⋊S3, C5⋊D4, C3×C15, C3×Dic5, Dic15, C6×D5, S3×C10, D30, C3⋊D12, C3×D15, C5×C3⋊S3, C3×C30, C15⋊D4, C5⋊D12, C3×Dic15, C6×D15, C10×C3⋊S3, D30⋊S3
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, D12, C3⋊D4, S32, C5⋊D4, S3×D5, C3⋊D12, C15⋊D4, C5⋊D12, D15⋊S3, D30⋊S3

Smallest permutation representation of D30⋊S3
►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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C 4 5A5B6A6B6C6D6E10A10B10C10D10E10F12A12B15A···15H30A···30H
order122233345566666101010101010121215···1530···30
size11183022430222243030221818181830304···44···4

39 irreducible representations

dim11112222222224444444
type++++++++++++++-+
imageC1C2C2C2S3S3D4D5D6D10D12C3⋊D4C5⋊D4S32S3×D5C3⋊D12C15⋊D4C5⋊D12D15⋊S3D30⋊S3
kernelD30⋊S3C3×Dic15C6×D15C10×C3⋊S3Dic15D30C3×C15C2×C3⋊S3C30C3×C6C15C15C32C10C6C5C3C3C2C1
# reps11111112222241412244

Matrix representation of D30⋊S3 ►in GL8(𝔽61)

4360000000
10000000
006000000
000600000
00001000
00000100
000000060
000000160
,
5623000000
525000000
000600000
006000000
000060000
000006000
000000160
000000060
,
10000000
01000000
00100000
00010000
000060100
000060000
00000010
00000001
,
600000000
060000000
00100000
000600000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(61))| [43,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60],[56,52,0,0,0,0,0,0,23,5,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

D30⋊S3 in GAP, Magma, Sage, TeX 

D_{30}\rtimes S_3
 
% in TeX
 
G:=Group("D30:S3");
 
// GroupNames label
 
G:=SmallGroup(360,86);
 
// by ID
 
G=gap.SmallGroup(360,86);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,387,201,730,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^30=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^25*b,d*c*d=c^-1>;
 
// generators/relations
 

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