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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, (C3xC15):13D4, (C6xD15):7C2, C6.25(S3xD5), C5:3(C3:D12), C3:3(C5:D12), C3:1(C15:D4), C15:5(C3:D4), (C3xC6).10D10, C32:4(C5:D4), C2.3(D15:S3), (C3xDic15):10C2, (C3xC30).24C22, (C2xC3:S3):3D5, (C10xC3:S3):3C2, SmallGroup(360,86)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC30 — D30:S3
Chief series
C1 — C5 — C15 — C3xC15 — C3xC30 — C6xD15 — D30:S3
Lower central
C3xC15 — C3xC30 — 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, C2xC6, C15, C15, C3xS3, C3:S3, C3xC6, Dic5, D10, C2xC10, D12, C3:D4, C5xS3, C3xD5, D15, C30, C30, C3xDic3, S3xC6, C2xC3:S3, C5:D4, C3xC15, C3xDic5, Dic15, C6xD5, S3xC10, D30, C3:D12, C3xD15, C5xC3:S3, C3xC30, C15:D4, C5:D12, C3xDic15, C6xD15, C10xC3:S3, D30:S3
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, D12, C3:D4, S32, C5:D4, S3xD5, 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:D4S32S3xD5C3:D12C15:D4C5:D12D15:S3D30:S3
kernelD30:S3C3xDic15C6xD15C10xC3:S3Dic15D30C3xC15C2xC3:S3C30C3xC6C15C15C32C10C6C5C3C3C2C1
# 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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