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G = D62D15order 360 = 23·32·5

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

metabelian, supersoluble, monomial

Aliases: D62D15, C157D12, C6.6D30, C30.26D6, Dic151S3, 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), C52(C3⋊D12), C31(C157D4), C154(C3⋊D4), C32(C5⋊D12), C323(C5⋊D4), (C3×Dic15)⋊7C2, (C3×C30).20C22, (C2×C3⋊D15)⋊6C2, SmallGroup(360,82)

Series: Derived Chief Lower central Upper central

C1C3×C30 — D62D15
C1C5C15C3×C15C3×C30C3×Dic15 — D62D15
C3×C15C3×C30 — D62D15
C1C2

Generators and relations for D62D15
 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 [×2], C3 [×2], C3, C4, C22 [×2], C5, S3 [×5], C6 [×2], C6 [×2], D4, C32, D5, C10, C10, Dic3, C12, D6, D6 [×3], C2×C6, C15 [×2], C15, C3×S3, C3⋊S3, C3×C6, Dic5, D10, C2×C10, D12, C3⋊D4, C5×S3, D15 [×4], C30 [×2], C30 [×2], C3×Dic3, S3×C6, C2×C3⋊S3, C5⋊D4, C3×C15, C3×Dic5, Dic15, S3×C10, D30 [×3], C2×C30, C3⋊D12, S3×C15, C3⋊D15, C3×C30, C5⋊D12, C157D4, C3×Dic15, S3×C30, C2×C3⋊D15, D62D15
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D5, D6 [×2], D10, D12, C3⋊D4, D15, S32, C5⋊D4, S3×D5, D30, C3⋊D12, C5⋊D12, C157D4, S3×D15, D62D15

Smallest permutation representation of D62D15
On 60 points
Generators in S60
(1 26 6 16 11 21)(2 27 7 17 12 22)(3 28 8 18 13 23)(4 29 9 19 14 24)(5 30 10 20 15 25)(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 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 31)(29 32)(30 33)
(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 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 50)(32 49)(33 48)(34 47)(35 46)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)

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

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

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

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⋊D4D30C157D4S32S3×D5C3⋊D12C5⋊D12S3×D15D62D15
kernelD62D15C3×Dic15S3×C30C2×C3⋊D15Dic15S3×C10C3×C15S3×C6C30C3×C6C15C15D6C32C6C3C10C6C5C3C2C1
# reps1111111222224448121244

Matrix representation of D62D15 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] >;

D62D15 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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