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G = D6⋊2D15  order 360 = 23·32·5

2nd semidirect product of D6 and D15 acting via D15/C15=C2

metabelian, supersoluble, monomial

Aliases: D6⋊2D15, C15⋊7D12, C6.6D30, C30.26D6, Dic15⋊1S3, C10.6S32, (S3×C6)⋊4D5, (C3×C15)⋊12D4, C6.6(S3×D5), (S3×C10)⋊2S3, (S3×C30)⋊4C2, (C3×C6).6D10, C2.6(S3×D15), C5⋊2(C3⋊D12), C3⋊1(C15⋊7D4), C15⋊4(C3⋊D4), C3⋊2(C5⋊D12), C32⋊3(C5⋊D4), (C3×Dic15)⋊7C2, (C3×C30).20C22, (C2×C3⋊D15)⋊6C2, SmallGroup(360,82)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C30 — D6⋊2D15
Chief series
C1 — C5 — C15 — C3×C15 — C3×C30 — C3×Dic15 — D6⋊2D15
Lower central
C3×C15 — C3×C30 — D6⋊2D15
Upper central
C1 — C2

Generators and relations for D6⋊2D15
 G = < a,b,c,d | a6=b2=c15=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

Subgroups: 612 in 74 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C15, C3×S3, C3⋊S3, C3×C6, Dic5, D10, C2×C10, D12, C3⋊D4, C5×S3, D15, C30, C30, C3×Dic3, S3×C6, C2×C3⋊S3, C5⋊D4, C3×C15, C3×Dic5, Dic15, S3×C10, D30, C2×C30, C3⋊D12, S3×C15, C3⋊D15, C3×C30, C5⋊D12, C15⋊7D4, C3×Dic15, S3×C30, C2×C3⋊D15, D6⋊2D15
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, D12, C3⋊D4, D15, S32, C5⋊D4, S3×D5, D30, C3⋊D12, C5⋊D12, C15⋊7D4, S3×D15, D6⋊2D15

Smallest permutation representation of D6⋊2D15
►On 60 points
Generators in S60
(1 30 6 20 11 25)(2 16 7 21 12 26)(3 17 8 22 13 27)(4 18 9 23 14 28)(5 19 10 24 15 29)(31 55 41 50 36 60)(32 56 42 51 37 46)(33 57 43 52 38 47)(34 58 44 53 39 48)(35 59 45 54 40 49)

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

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

51 conjugacy classes

class 1 2A2B2C3A3B3C 4 5A5B6A6B6C6D6E10A10B10C10D10E10F12A12B15A15B15C15D15E···15J30A30B30C30D30E···30J30K···30R
order12223334556666610101010101012121515151515···153030303030···3030···30
size11690224302222466226666303022224···422224···46···6

51 irreducible representations

dim1111222222222222444444
type+++++++++++++++++++
imageC1C2C2C2S3S3D4D5D6D10D12C3⋊D4D15C5⋊D4D30C15⋊7D4S32S3×D5C3⋊D12C5⋊D12S3×D15D6⋊2D15
kernelD6⋊2D15C3×Dic15S3×C30C2×C3⋊D15Dic15S3×C10C3×C15S3×C6C30C3×C6C15C15D6C32C6C3C10C6C5C3C2C1
# reps1111111222224448121244

Matrix representation of D6⋊2D15 ►in GL8(𝔽61)

10000000
01000000
006000000
000600000
00001000
00000100
000000601
000000600
,
600000000
060000000
000510000
00600000
000060000
000006000
000000600
000000601
,
1843000000
1860000000
00100000
00010000
000006000
000016000
00000010
00000001
,
01000000
10000000
00100000
000600000
000016000
000006000
00000001
00000010

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1],[18,18,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

D6⋊2D15 in GAP, Magma, Sage, TeX 

D_6\rtimes_2D_{15}
 
% in TeX
 
G:=Group("D6:2D15");
 
// GroupNames label
 
G:=SmallGroup(360,82);
 
// by ID
 
G=gap.SmallGroup(360,82);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,201,1444,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^15=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

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