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## G = D15⋊SD16order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D15⋊SD16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C20⋊D6 — D15⋊SD16
 Lower central C15 — C30 — C60 — D15⋊SD16
 Upper central C1 — C2 — C4 — Q8

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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 20A 20B 20C 20D 24A 24B 30A 30B 40A 40B 40C 40D 60A ··· 60F order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 8 8 8 8 10 10 10 10 12 12 15 15 20 20 20 20 24 24 30 30 40 40 40 40 60 ··· 60 size 1 1 12 15 15 20 2 2 4 30 60 2 2 2 40 6 6 10 10 2 2 24 24 4 8 4 4 4 4 8 8 20 20 4 4 12 12 12 12 8 ··· 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 S3×D4 S3×D5 D4×D5 S3×SD16 C2×S3×D5 D5×SD16 D10⋊D6 D15⋊SD16 kernel D15⋊SD16 D15⋊2C8 C6.D20 D12.D5 C3×Q8⋊D5 C5×Q8⋊2S3 C20⋊D6 Q8×D15 Q8⋊D5 Dic15 D30 Q8⋊2S3 C5⋊2C8 D20 C5×Q8 D15 C3⋊C8 D12 C3×Q8 C10 Q8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of D15⋊SD16 in GL6(𝔽241)

 240 189 0 0 0 0 52 52 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0
,
 240 189 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 240 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 38 73 0 0 0 0 208 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 0 240 123 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 240 0

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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