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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 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8 [×2], C23, D5 [×3], C10, C10, Dic3 [×2], C12, C12, D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20, C20, D10 [×4], C2×C10, C3⋊C8, C24, Dic6 [×2], C4×S3 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C2×SD16, C52C8, C40, Dic10 [×2], C4×D5 [×2], 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 [×2], 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 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], 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)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(76 84)(77 83)(78 82)(79 81)(85 90)(86 89)(87 88)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
(1 106 43 88 24 91 46 69)(2 117 44 84 25 102 47 65)(3 113 45 80 26 98 48 61)(4 109 31 76 27 94 49 72)(5 120 32 87 28 105 50 68)(6 116 33 83 29 101 51 64)(7 112 34 79 30 97 52 75)(8 108 35 90 16 93 53 71)(9 119 36 86 17 104 54 67)(10 115 37 82 18 100 55 63)(11 111 38 78 19 96 56 74)(12 107 39 89 20 92 57 70)(13 118 40 85 21 103 58 66)(14 114 41 81 22 99 59 62)(15 110 42 77 23 95 60 73)
(2 12)(3 8)(5 15)(6 11)(9 14)(16 26)(17 22)(19 29)(20 25)(23 28)(31 49)(32 60)(33 56)(34 52)(35 48)(36 59)(37 55)(38 51)(39 47)(40 58)(41 54)(42 50)(43 46)(44 57)(45 53)(61 93)(62 104)(63 100)(64 96)(65 92)(66 103)(67 99)(68 95)(69 91)(70 102)(71 98)(72 94)(73 105)(74 101)(75 97)(76 109)(77 120)(78 116)(79 112)(80 108)(81 119)(82 115)(83 111)(84 107)(85 118)(86 114)(87 110)(88 106)(89 117)(90 113)

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

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

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

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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