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G = D8⋊D15order 480 = 25·3·5

2nd semidirect product of D8 and D15 acting via D15/C15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C82D30, D82D15, D42D30, C4012D6, C2412D10, C1209C22, D30.30D4, C60.65C23, Dic15.35D4, D60.20C22, Dic3012C22, (C5×D4)⋊6D6, (C5×D8)⋊4S3, (C3×D8)⋊4D5, (D4×D15)⋊9C2, (C15×D8)⋊4C2, (C3×D4)⋊6D10, C55(D8⋊S3), C24⋊D55C2, C40⋊S33C2, D4⋊D1510C2, C35(D8⋊D5), D4.D159C2, C6.109(D4×D5), C2.16(D4×D15), D42D158C2, C1528(C8⋊C22), C10.111(S3×D4), C30.316(C2×D4), C4.2(C22×D15), C153C815C22, (D4×C15)⋊15C22, C20.103(C22×S3), (C4×D15).24C22, C12.103(C22×D5), SmallGroup(480,876)

Series: Derived Chief Lower central Upper central

C1C60 — D8⋊D15
C1C5C15C30C60C4×D15D4×D15 — D8⋊D15
C15C30C60 — D8⋊D15
C1C2C4D8

Generators and relations for D8⋊D15
 G = < a,b,c,d | a4=b2=c30=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c-1 >

Subgroups: 1012 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, D15 [×2], C30, C30 [×2], C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, Dic15, Dic15, C60, D30, D30 [×3], C2×C30 [×2], C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D8⋊S3, C153C8, C120, Dic30, C4×D15, D60, C2×Dic15, C157D4 [×2], D4×C15 [×2], C22×D15, D8⋊D5, C40⋊S3, C24⋊D5, D4⋊D15, D4.D15, C15×D8, D4×D15, D42D15, D8⋊D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C8⋊C22, C22×D5, S3×D4, D30 [×3], D4×D5, D8⋊S3, C22×D15, D8⋊D5, D4×D15, D8⋊D15

Smallest permutation representation of D8⋊D15
On 120 points
Generators in S120
(1 73 19 88)(2 89 20 74)(3 75 21 90)(4 61 22 76)(5 77 23 62)(6 63 24 78)(7 79 25 64)(8 65 26 80)(9 81 27 66)(10 67 28 82)(11 83 29 68)(12 69 30 84)(13 85 16 70)(14 71 17 86)(15 87 18 72)(31 46 115 100)(32 101 116 47)(33 48 117 102)(34 103 118 49)(35 50 119 104)(36 105 120 51)(37 52 91 106)(38 107 92 53)(39 54 93 108)(40 109 94 55)(41 56 95 110)(42 111 96 57)(43 58 97 112)(44 113 98 59)(45 60 99 114)
(1 92)(2 108)(3 94)(4 110)(5 96)(6 112)(7 98)(8 114)(9 100)(10 116)(11 102)(12 118)(13 104)(14 120)(15 106)(16 50)(17 36)(18 52)(19 38)(20 54)(21 40)(22 56)(23 42)(24 58)(25 44)(26 60)(27 46)(28 32)(29 48)(30 34)(31 66)(33 68)(35 70)(37 72)(39 74)(41 76)(43 78)(45 80)(47 82)(49 84)(51 86)(53 88)(55 90)(57 62)(59 64)(61 95)(63 97)(65 99)(67 101)(69 103)(71 105)(73 107)(75 109)(77 111)(79 113)(81 115)(83 117)(85 119)(87 91)(89 93)
(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)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(15 30)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(119 120)

G:=sub<Sym(120)| (1,73,19,88)(2,89,20,74)(3,75,21,90)(4,61,22,76)(5,77,23,62)(6,63,24,78)(7,79,25,64)(8,65,26,80)(9,81,27,66)(10,67,28,82)(11,83,29,68)(12,69,30,84)(13,85,16,70)(14,71,17,86)(15,87,18,72)(31,46,115,100)(32,101,116,47)(33,48,117,102)(34,103,118,49)(35,50,119,104)(36,105,120,51)(37,52,91,106)(38,107,92,53)(39,54,93,108)(40,109,94,55)(41,56,95,110)(42,111,96,57)(43,58,97,112)(44,113,98,59)(45,60,99,114), (1,92)(2,108)(3,94)(4,110)(5,96)(6,112)(7,98)(8,114)(9,100)(10,116)(11,102)(12,118)(13,104)(14,120)(15,106)(16,50)(17,36)(18,52)(19,38)(20,54)(21,40)(22,56)(23,42)(24,58)(25,44)(26,60)(27,46)(28,32)(29,48)(30,34)(31,66)(33,68)(35,70)(37,72)(39,74)(41,76)(43,78)(45,80)(47,82)(49,84)(51,86)(53,88)(55,90)(57,62)(59,64)(61,95)(63,97)(65,99)(67,101)(69,103)(71,105)(73,107)(75,109)(77,111)(79,113)(81,115)(83,117)(85,119)(87,91)(89,93), (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(15,30)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(119,120)>;

G:=Group( (1,73,19,88)(2,89,20,74)(3,75,21,90)(4,61,22,76)(5,77,23,62)(6,63,24,78)(7,79,25,64)(8,65,26,80)(9,81,27,66)(10,67,28,82)(11,83,29,68)(12,69,30,84)(13,85,16,70)(14,71,17,86)(15,87,18,72)(31,46,115,100)(32,101,116,47)(33,48,117,102)(34,103,118,49)(35,50,119,104)(36,105,120,51)(37,52,91,106)(38,107,92,53)(39,54,93,108)(40,109,94,55)(41,56,95,110)(42,111,96,57)(43,58,97,112)(44,113,98,59)(45,60,99,114), (1,92)(2,108)(3,94)(4,110)(5,96)(6,112)(7,98)(8,114)(9,100)(10,116)(11,102)(12,118)(13,104)(14,120)(15,106)(16,50)(17,36)(18,52)(19,38)(20,54)(21,40)(22,56)(23,42)(24,58)(25,44)(26,60)(27,46)(28,32)(29,48)(30,34)(31,66)(33,68)(35,70)(37,72)(39,74)(41,76)(43,78)(45,80)(47,82)(49,84)(51,86)(53,88)(55,90)(57,62)(59,64)(61,95)(63,97)(65,99)(67,101)(69,103)(71,105)(73,107)(75,109)(77,111)(79,113)(81,115)(83,117)(85,119)(87,91)(89,93), (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(15,30)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(119,120) );

G=PermutationGroup([(1,73,19,88),(2,89,20,74),(3,75,21,90),(4,61,22,76),(5,77,23,62),(6,63,24,78),(7,79,25,64),(8,65,26,80),(9,81,27,66),(10,67,28,82),(11,83,29,68),(12,69,30,84),(13,85,16,70),(14,71,17,86),(15,87,18,72),(31,46,115,100),(32,101,116,47),(33,48,117,102),(34,103,118,49),(35,50,119,104),(36,105,120,51),(37,52,91,106),(38,107,92,53),(39,54,93,108),(40,109,94,55),(41,56,95,110),(42,111,96,57),(43,58,97,112),(44,113,98,59),(45,60,99,114)], [(1,92),(2,108),(3,94),(4,110),(5,96),(6,112),(7,98),(8,114),(9,100),(10,116),(11,102),(12,118),(13,104),(14,120),(15,106),(16,50),(17,36),(18,52),(19,38),(20,54),(21,40),(22,56),(23,42),(24,58),(25,44),(26,60),(27,46),(28,32),(29,48),(30,34),(31,66),(33,68),(35,70),(37,72),(39,74),(41,76),(43,78),(45,80),(47,82),(49,84),(51,86),(53,88),(55,90),(57,62),(59,64),(61,95),(63,97),(65,99),(67,101),(69,103),(71,105),(73,107),(75,109),(77,111),(79,113),(81,115),(83,117),(85,119),(87,91),(89,93)], [(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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(15,30),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,65),(62,64),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(91,118),(92,117),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105),(119,120)])

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D10E10F 12 15A15B15C15D20A20B24A24B30A30B30C30D30E···30L40A40B40C40D60A60B60C60D120A···120H
order122222344455666881010101010101215151515202024243030303030···304040404060606060120···120
size114430602230602228846022888842222444422228···8444444444···4

60 irreducible representations

dim11111111222222222224444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D10D10D15D30D30C8⋊C22S3×D4D4×D5D8⋊S3D8⋊D5D4×D15D8⋊D15
kernelD8⋊D15C40⋊S3C24⋊D5D4⋊D15D4.D15C15×D8D4×D15D42D15C5×D8Dic15D30C3×D8C40C5×D4C24C3×D4D8C8D4C15C10C6C5C3C2C1
# reps11111111111212244481122448

Matrix representation of D8⋊D15 in GL6(𝔽241)

100000
010000
00102390
00010239
00102400
00010240
,
24000000
02400000
009453147188
001881475394
0047147147188
00941945394
,
522400000
532400000
0024012239
00240020
00001240
000010
,
0510000
5200000
00240020
0024012239
000010
00001240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,239,0,240,0,0,0,0,239,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,94,188,47,94,0,0,53,147,147,194,0,0,147,53,147,53,0,0,188,94,188,94],[52,53,0,0,0,0,240,240,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,2,2,1,1,0,0,239,0,240,0],[0,52,0,0,0,0,51,0,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,2,2,1,1,0,0,0,239,0,240] >;

D8⋊D15 in GAP, Magma, Sage, TeX

D_8\rtimes D_{15}
% in TeX

G:=Group("D8:D15");
// GroupNames label

G:=SmallGroup(480,876);
// by ID

G=gap.SmallGroup(480,876);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^30=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

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