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G = D15⋊SD16order 480 = 25·3·5

The semidirect product of D15 and SD16 acting via SD16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D153SD16, D20.11D6, D30.41D4, D12.11D10, C60.29C23, Dic15.15D4, Dic306C22, C3⋊C818D10, Q8⋊D52S3, (C5×Q8)⋊6D6, Q86(S3×D5), C34(D5×SD16), C54(S3×SD16), C52C818D6, (Q8×D15)⋊1C2, (C3×Q8)⋊3D10, C6.75(D4×D5), Q82S32D5, C10.76(S3×D4), C1515(C2×SD16), D152C85C2, D12.D55C2, C30.191(C2×D4), C6.D205C2, C20⋊D6.1C2, (Q8×C15)⋊5C22, C20.29(C22×S3), (C4×D15).9C22, C12.29(C22×D5), (C5×D12).10C22, (C3×D20).10C22, C2.28(D10⋊D6), C4.29(C2×S3×D5), (C3×Q8⋊D5)⋊3C2, (C5×C3⋊C8)⋊13C22, (C5×Q82S3)⋊3C2, (C3×C52C8)⋊13C22, SmallGroup(480,581)

Series: Derived Chief Lower central Upper central

C1C60 — D15⋊SD16
C1C5C15C30C60C3×D20C20⋊D6 — D15⋊SD16
C15C30C60 — D15⋊SD16
C1C2C4Q8

Generators and relations for D15⋊SD16
 G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c3 >

Subgroups: 924 in 136 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, Dic15, Dic15, C60, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, S3×SD16, C5×C3⋊C8, C3×C52C8, C15⋊D4, C3×D20, C5×D12, Dic30, Dic30, C4×D15, C4×D15, Q8×C15, C2×S3×D5, D5×SD16, D152C8, C6.D20, D12.D5, C3×Q8⋊D5, C5×Q82S3, C20⋊D6, Q8×D15, D15⋊SD16
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, C2×SD16, C22×D5, S3×D4, S3×D5, D4×D5, S3×SD16, C2×S3×D5, D5×SD16, D10⋊D6, D15⋊SD16

Smallest permutation representation of D15⋊SD16
On 120 points
Generators in S120
(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 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 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 99)(92 98)(93 97)(94 96)(100 105)(101 104)(102 103)(106 114)(107 113)(108 112)(109 111)(115 120)(116 119)(117 118)
(1 118 43 76 16 103 54 61)(2 114 44 87 17 99 55 72)(3 110 45 83 18 95 56 68)(4 106 31 79 19 91 57 64)(5 117 32 90 20 102 58 75)(6 113 33 86 21 98 59 71)(7 109 34 82 22 94 60 67)(8 120 35 78 23 105 46 63)(9 116 36 89 24 101 47 74)(10 112 37 85 25 97 48 70)(11 108 38 81 26 93 49 66)(12 119 39 77 27 104 50 62)(13 115 40 88 28 100 51 73)(14 111 41 84 29 96 52 69)(15 107 42 80 30 92 53 65)
(2 12)(3 8)(5 15)(6 11)(9 14)(17 27)(18 23)(20 30)(21 26)(24 29)(31 57)(32 53)(33 49)(34 60)(35 56)(36 52)(37 48)(38 59)(39 55)(40 51)(41 47)(42 58)(43 54)(44 50)(45 46)(61 103)(62 99)(63 95)(64 91)(65 102)(66 98)(67 94)(68 105)(69 101)(70 97)(71 93)(72 104)(73 100)(74 96)(75 92)(76 118)(77 114)(78 110)(79 106)(80 117)(81 113)(82 109)(83 120)(84 116)(85 112)(86 108)(87 119)(88 115)(89 111)(90 107)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B8A8B8C8D10A10B10C10D12A12B15A15B20A20B20C20D24A24B30A30B40A40B40C40D60A···60F
order1222223444455668888101010101212151520202020242430304040404060···60
size111215152022430602224066101022242448444488202044121212128···8

45 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6SD16D10D10D10S3×D4S3×D5D4×D5S3×SD16C2×S3×D5D5×SD16D10⋊D6D15⋊SD16
kernelD15⋊SD16D152C8C6.D20D12.D5C3×Q8⋊D5C5×Q82S3C20⋊D6Q8×D15Q8⋊D5Dic15D30Q82S3C52C8D20C5×Q8D15C3⋊C8D12C3×Q8C10Q8C6C5C4C3C2C1
# reps111111111112111422212222442

Matrix representation of D15⋊SD16 in GL6(𝔽241)

2401890000
52520000
001000
000100
00002401
00002400
,
2401890000
010000
00240000
00024000
00002400
00002401
,
100000
010000
00387300
00208000
000001
000010
,
100000
010000
0024012300
000100
00000240
00002400

G:=sub<GL(6,GF(241))| [240,52,0,0,0,0,189,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,0,0,0,0,0,189,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,240,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,38,208,0,0,0,0,73,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,123,1,0,0,0,0,0,0,0,240,0,0,0,0,240,0] >;

D15⋊SD16 in GAP, Magma, Sage, TeX

D_{15}\rtimes {\rm SD}_{16}
% in TeX

G:=Group("D15:SD16");
// GroupNames label

G:=SmallGroup(480,581);
// by ID

G=gap.SmallGroup(480,581);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,100,675,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^3>;
// generators/relations

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