metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊4D6, D15⋊2D8, D12⋊4D10, D30.37D4, D60⋊3C22, C60.5C23, Dic15.11D4, C5⋊3(S3×D8), C3⋊3(D5×D8), C15⋊9(C2×D8), D4⋊S3⋊2D5, D4⋊D5⋊2S3, C3⋊C8⋊14D10, (C5×D4)⋊3D6, D4⋊5(S3×D5), (C3×D4)⋊3D10, (D4×D15)⋊1C2, C3⋊D40⋊2C2, C5⋊2C8⋊14D6, C5⋊D24⋊2C2, C6.67(D4×D5), C20⋊D6⋊1C2, C10.68(S3×D4), D15⋊2C8⋊1C2, C30.167(C2×D4), (D4×C15)⋊5C22, (C5×D12)⋊3C22, (C3×D20)⋊3C22, C20.5(C22×S3), C12.5(C22×D5), (C4×D15).1C22, C2.20(D10⋊D6), C4.5(C2×S3×D5), (C5×D4⋊S3)⋊3C2, (C3×D4⋊D5)⋊3C2, (C5×C3⋊C8)⋊3C22, (C3×C5⋊2C8)⋊3C22, SmallGroup(480,557)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D15⋊D8
G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c-1 >
Subgroups: 1228 in 152 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, D15, C30, C30, C2×D8, C5⋊2C8, C40, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×D4, C22×D5, S3×C8, D24, D4⋊S3, D4⋊S3, C3×D8, S3×D4, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D30, C2×C30, C8×D5, D40, D4⋊D5, D4⋊D5, C5×D8, D4×D5, S3×D8, C5×C3⋊C8, C3×C5⋊2C8, C15⋊D4, C3×D20, C5×D12, C4×D15, D60, C15⋊7D4, D4×C15, C2×S3×D5, C22×D15, D5×D8, D15⋊2C8, C3⋊D40, C5⋊D24, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D4×D15, D15⋊D8
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, C2×D8, C22×D5, S3×D4, S3×D5, D4×D5, S3×D8, C2×S3×D5, D5×D8, D10⋊D6, D15⋊D8
►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 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(46 53)(47 52)(48 51)(49 50)(54 60)(55 59)(56 58)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 92)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(106 114)(107 113)(108 112)(109 111)(115 120)(116 119)(117 118)
(1 62 31 118 16 84 50 92)(2 73 32 114 17 80 51 103)(3 69 33 110 18 76 52 99)(4 65 34 106 19 87 53 95)(5 61 35 117 20 83 54 91)(6 72 36 113 21 79 55 102)(7 68 37 109 22 90 56 98)(8 64 38 120 23 86 57 94)(9 75 39 116 24 82 58 105)(10 71 40 112 25 78 59 101)(11 67 41 108 26 89 60 97)(12 63 42 119 27 85 46 93)(13 74 43 115 28 81 47 104)(14 70 44 111 29 77 48 100)(15 66 45 107 30 88 49 96)
(2 12)(3 8)(5 15)(6 11)(9 14)(17 27)(18 23)(20 30)(21 26)(24 29)(31 50)(32 46)(33 57)(34 53)(35 49)(36 60)(37 56)(38 52)(39 48)(40 59)(41 55)(42 51)(43 47)(44 58)(45 54)(61 96)(62 92)(63 103)(64 99)(65 95)(66 91)(67 102)(68 98)(69 94)(70 105)(71 101)(72 97)(73 93)(74 104)(75 100)(76 120)(77 116)(78 112)(79 108)(80 119)(81 115)(82 111)(83 107)(84 118)(85 114)(86 110)(87 106)(88 117)(89 113)(90 109)
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)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(46,53)(47,52)(48,51)(49,50)(54,60)(55,59)(56,58)(61,62)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,92)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,114)(107,113)(108,112)(109,111)(115,120)(116,119)(117,118), (1,62,31,118,16,84,50,92)(2,73,32,114,17,80,51,103)(3,69,33,110,18,76,52,99)(4,65,34,106,19,87,53,95)(5,61,35,117,20,83,54,91)(6,72,36,113,21,79,55,102)(7,68,37,109,22,90,56,98)(8,64,38,120,23,86,57,94)(9,75,39,116,24,82,58,105)(10,71,40,112,25,78,59,101)(11,67,41,108,26,89,60,97)(12,63,42,119,27,85,46,93)(13,74,43,115,28,81,47,104)(14,70,44,111,29,77,48,100)(15,66,45,107,30,88,49,96), (2,12)(3,8)(5,15)(6,11)(9,14)(17,27)(18,23)(20,30)(21,26)(24,29)(31,50)(32,46)(33,57)(34,53)(35,49)(36,60)(37,56)(38,52)(39,48)(40,59)(41,55)(42,51)(43,47)(44,58)(45,54)(61,96)(62,92)(63,103)(64,99)(65,95)(66,91)(67,102)(68,98)(69,94)(70,105)(71,101)(72,97)(73,93)(74,104)(75,100)(76,120)(77,116)(78,112)(79,108)(80,119)(81,115)(82,111)(83,107)(84,118)(85,114)(86,110)(87,106)(88,117)(89,113)(90,109)>;
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)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(46,53)(47,52)(48,51)(49,50)(54,60)(55,59)(56,58)(61,62)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,92)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(106,114)(107,113)(108,112)(109,111)(115,120)(116,119)(117,118), (1,62,31,118,16,84,50,92)(2,73,32,114,17,80,51,103)(3,69,33,110,18,76,52,99)(4,65,34,106,19,87,53,95)(5,61,35,117,20,83,54,91)(6,72,36,113,21,79,55,102)(7,68,37,109,22,90,56,98)(8,64,38,120,23,86,57,94)(9,75,39,116,24,82,58,105)(10,71,40,112,25,78,59,101)(11,67,41,108,26,89,60,97)(12,63,42,119,27,85,46,93)(13,74,43,115,28,81,47,104)(14,70,44,111,29,77,48,100)(15,66,45,107,30,88,49,96), (2,12)(3,8)(5,15)(6,11)(9,14)(17,27)(18,23)(20,30)(21,26)(24,29)(31,50)(32,46)(33,57)(34,53)(35,49)(36,60)(37,56)(38,52)(39,48)(40,59)(41,55)(42,51)(43,47)(44,58)(45,54)(61,96)(62,92)(63,103)(64,99)(65,95)(66,91)(67,102)(68,98)(69,94)(70,105)(71,101)(72,97)(73,93)(74,104)(75,100)(76,120)(77,116)(78,112)(79,108)(80,119)(81,115)(82,111)(83,107)(84,118)(85,114)(86,110)(87,106)(88,117)(89,113)(90,109) );
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),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(46,53),(47,52),(48,51),(49,50),(54,60),(55,59),(56,58),(61,62),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(91,92),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(106,114),(107,113),(108,112),(109,111),(115,120),(116,119),(117,118)], [(1,62,31,118,16,84,50,92),(2,73,32,114,17,80,51,103),(3,69,33,110,18,76,52,99),(4,65,34,106,19,87,53,95),(5,61,35,117,20,83,54,91),(6,72,36,113,21,79,55,102),(7,68,37,109,22,90,56,98),(8,64,38,120,23,86,57,94),(9,75,39,116,24,82,58,105),(10,71,40,112,25,78,59,101),(11,67,41,108,26,89,60,97),(12,63,42,119,27,85,46,93),(13,74,43,115,28,81,47,104),(14,70,44,111,29,77,48,100),(15,66,45,107,30,88,49,96)], [(2,12),(3,8),(5,15),(6,11),(9,14),(17,27),(18,23),(20,30),(21,26),(24,29),(31,50),(32,46),(33,57),(34,53),(35,49),(36,60),(37,56),(38,52),(39,48),(40,59),(41,55),(42,51),(43,47),(44,58),(45,54),(61,96),(62,92),(63,103),(64,99),(65,95),(66,91),(67,102),(68,98),(69,94),(70,105),(71,101),(72,97),(73,93),(74,104),(75,100),(76,120),(77,116),(78,112),(79,108),(80,119),(81,115),(82,111),(83,107),(84,118),(85,114),(86,110),(87,106),(88,117),(89,113),(90,109)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 12 | 15 | 15 | 20 | 60 | 2 | 2 | 30 | 2 | 2 | 2 | 8 | 40 | 6 | 6 | 10 | 10 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | D8 | D10 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | S3×D8 | C2×S3×D5 | D5×D8 | D10⋊D6 | D15⋊D8 |
| kernel | D15⋊D8 | D15⋊2C8 | C3⋊D40 | C5⋊D24 | C3×D4⋊D5 | C5×D4⋊S3 | C20⋊D6 | D4×D15 | D4⋊D5 | Dic15 | D30 | D4⋊S3 | C5⋊2C8 | D20 | C5×D4 | D15 | C3⋊C8 | D12 | C3×D4 | C10 | D4 | 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⋊D8 ►in GL6(𝔽241)
| 240 | 189 | 0 | 0 | 0 | 0 |
| 52 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 54 | 0 | 0 |
| 0 | 0 | 174 | 239 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 52 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 187 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 174 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 230 | 22 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 174 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
G:=sub<GL(6,GF(241))| [240,52,0,0,0,0,189,52,0,0,0,0,0,0,1,174,0,0,0,0,54,239,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,52,240,0,0,0,0,0,0,240,0,0,0,0,0,187,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,0,230,0,0,0,0,22,22],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,240] >;
D15⋊D8 in GAP, Magma, Sage, TeX
D_{15}\rtimes D_8 % in TeX
G:=Group("D15:D8"); // GroupNames label
G:=SmallGroup(480,557);
// by ID
G=gap.SmallGroup(480,557);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,675,346,185,80,1356,18822]);
// 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^-1>;
// generators/relations