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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 [×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 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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