metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D15⋊3SD16, D20.11D6, D30.41D4, D12.11D10, C60.29C23, Dic15.15D4, Dic30⋊6C22, C3⋊C8⋊18D10, Q8⋊D5⋊2S3, (C5×Q8)⋊6D6, Q8⋊6(S3×D5), C3⋊4(D5×SD16), C5⋊4(S3×SD16), C5⋊2C8⋊18D6, (Q8×D15)⋊1C2, (C3×Q8)⋊3D10, C6.75(D4×D5), Q8⋊2S3⋊2D5, C10.76(S3×D4), C15⋊15(C2×SD16), D15⋊2C8⋊5C2, D12.D5⋊5C2, C30.191(C2×D4), C6.D20⋊5C2, 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×Q8⋊2S3)⋊3C2, (C3×C5⋊2C8)⋊13C22, SmallGroup(480,581)
Series: Derived ►Chief ►Lower central ►Upper central
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, C5⋊2C8, C40, Dic10 [×2], C4×D5 [×2], D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, 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×C5⋊2C8, C15⋊D4, C3×D20, C5×D12, Dic30, Dic30, C4×D15, C4×D15, Q8×C15, C2×S3×D5, D5×SD16, D15⋊2C8, C6.D20, D12.D5, C3×Q8⋊D5, C5×Q8⋊2S3, 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
►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