metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D30, D8⋊2D15, D4⋊2D30, C40⋊12D6, C24⋊12D10, C120⋊9C22, D30.30D4, C60.65C23, Dic15.35D4, D60.20C22, Dic30⋊12C22, (C5×D4)⋊6D6, (C5×D8)⋊4S3, (C3×D8)⋊4D5, (D4×D15)⋊9C2, (C15×D8)⋊4C2, (C3×D4)⋊6D10, C5⋊5(D8⋊S3), C24⋊D5⋊5C2, C40⋊S3⋊3C2, D4⋊D15⋊10C2, C3⋊5(D8⋊D5), D4.D15⋊9C2, C6.109(D4×D5), C2.16(D4×D15), D4⋊2D15⋊8C2, C15⋊28(C8⋊C22), C10.111(S3×D4), C30.316(C2×D4), C4.2(C22×D15), C15⋊3C8⋊15C22, (D4×C15)⋊15C22, C20.103(C22×S3), (C4×D15).24C22, C12.103(C22×D5), SmallGroup(480,876)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊D15
G = < a,b,c,d | a4=b2=c30=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c-1 >
Subgroups: 1012 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, Dic15, Dic15, C60, D30, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D8⋊S3, C15⋊3C8, C120, Dic30, C4×D15, D60, C2×Dic15, C15⋊7D4, D4×C15, C22×D15, D8⋊D5, C40⋊S3, C24⋊D5, D4⋊D15, D4.D15, C15×D8, D4×D15, D4⋊2D15, D8⋊D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D15, C8⋊C22, C22×D5, S3×D4, D30, D4×D5, D8⋊S3, C22×D15, D8⋊D5, D4×D15, D8⋊D15
►On 120 points
Generators in S120
(1 36 30 51)(2 52 16 37)(3 38 17 53)(4 54 18 39)(5 40 19 55)(6 56 20 41)(7 42 21 57)(8 58 22 43)(9 44 23 59)(10 60 24 45)(11 46 25 31)(12 32 26 47)(13 48 27 33)(14 34 28 49)(15 50 29 35)(61 113 98 76)(62 77 99 114)(63 115 100 78)(64 79 101 116)(65 117 102 80)(66 81 103 118)(67 119 104 82)(68 83 105 120)(69 91 106 84)(70 85 107 92)(71 93 108 86)(72 87 109 94)(73 95 110 88)(74 89 111 96)(75 97 112 90)
(1 108)(2 94)(3 110)(4 96)(5 112)(6 98)(7 114)(8 100)(9 116)(10 102)(11 118)(12 104)(13 120)(14 106)(15 92)(16 87)(17 73)(18 89)(19 75)(20 61)(21 77)(22 63)(23 79)(24 65)(25 81)(26 67)(27 83)(28 69)(29 85)(30 71)(31 66)(32 119)(33 68)(34 91)(35 70)(36 93)(37 72)(38 95)(39 74)(40 97)(41 76)(42 99)(43 78)(44 101)(45 80)(46 103)(47 82)(48 105)(49 84)(50 107)(51 86)(52 109)(53 88)(54 111)(55 90)(56 113)(57 62)(58 115)(59 64)(60 117)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(15 30)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(56 60)(57 59)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 90)(82 89)(83 88)(84 87)(85 86)(91 94)(92 93)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(120)| (1,36,30,51)(2,52,16,37)(3,38,17,53)(4,54,18,39)(5,40,19,55)(6,56,20,41)(7,42,21,57)(8,58,22,43)(9,44,23,59)(10,60,24,45)(11,46,25,31)(12,32,26,47)(13,48,27,33)(14,34,28,49)(15,50,29,35)(61,113,98,76)(62,77,99,114)(63,115,100,78)(64,79,101,116)(65,117,102,80)(66,81,103,118)(67,119,104,82)(68,83,105,120)(69,91,106,84)(70,85,107,92)(71,93,108,86)(72,87,109,94)(73,95,110,88)(74,89,111,96)(75,97,112,90), (1,108)(2,94)(3,110)(4,96)(5,112)(6,98)(7,114)(8,100)(9,116)(10,102)(11,118)(12,104)(13,120)(14,106)(15,92)(16,87)(17,73)(18,89)(19,75)(20,61)(21,77)(22,63)(23,79)(24,65)(25,81)(26,67)(27,83)(28,69)(29,85)(30,71)(31,66)(32,119)(33,68)(34,91)(35,70)(36,93)(37,72)(38,95)(39,74)(40,97)(41,76)(42,99)(43,78)(44,101)(45,80)(46,103)(47,82)(48,105)(49,84)(50,107)(51,86)(52,109)(53,88)(54,111)(55,90)(56,113)(57,62)(58,115)(59,64)(60,117), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(15,30)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(91,94)(92,93)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,36,30,51)(2,52,16,37)(3,38,17,53)(4,54,18,39)(5,40,19,55)(6,56,20,41)(7,42,21,57)(8,58,22,43)(9,44,23,59)(10,60,24,45)(11,46,25,31)(12,32,26,47)(13,48,27,33)(14,34,28,49)(15,50,29,35)(61,113,98,76)(62,77,99,114)(63,115,100,78)(64,79,101,116)(65,117,102,80)(66,81,103,118)(67,119,104,82)(68,83,105,120)(69,91,106,84)(70,85,107,92)(71,93,108,86)(72,87,109,94)(73,95,110,88)(74,89,111,96)(75,97,112,90), (1,108)(2,94)(3,110)(4,96)(5,112)(6,98)(7,114)(8,100)(9,116)(10,102)(11,118)(12,104)(13,120)(14,106)(15,92)(16,87)(17,73)(18,89)(19,75)(20,61)(21,77)(22,63)(23,79)(24,65)(25,81)(26,67)(27,83)(28,69)(29,85)(30,71)(31,66)(32,119)(33,68)(34,91)(35,70)(36,93)(37,72)(38,95)(39,74)(40,97)(41,76)(42,99)(43,78)(44,101)(45,80)(46,103)(47,82)(48,105)(49,84)(50,107)(51,86)(52,109)(53,88)(54,111)(55,90)(56,113)(57,62)(58,115)(59,64)(60,117), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(15,30)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(91,94)(92,93)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([[(1,36,30,51),(2,52,16,37),(3,38,17,53),(4,54,18,39),(5,40,19,55),(6,56,20,41),(7,42,21,57),(8,58,22,43),(9,44,23,59),(10,60,24,45),(11,46,25,31),(12,32,26,47),(13,48,27,33),(14,34,28,49),(15,50,29,35),(61,113,98,76),(62,77,99,114),(63,115,100,78),(64,79,101,116),(65,117,102,80),(66,81,103,118),(67,119,104,82),(68,83,105,120),(69,91,106,84),(70,85,107,92),(71,93,108,86),(72,87,109,94),(73,95,110,88),(74,89,111,96),(75,97,112,90)], [(1,108),(2,94),(3,110),(4,96),(5,112),(6,98),(7,114),(8,100),(9,116),(10,102),(11,118),(12,104),(13,120),(14,106),(15,92),(16,87),(17,73),(18,89),(19,75),(20,61),(21,77),(22,63),(23,79),(24,65),(25,81),(26,67),(27,83),(28,69),(29,85),(30,71),(31,66),(32,119),(33,68),(34,91),(35,70),(36,93),(37,72),(38,95),(39,74),(40,97),(41,76),(42,99),(43,78),(44,101),(45,80),(46,103),(47,82),(48,105),(49,84),(50,107),(51,86),(52,109),(53,88),(54,111),(55,90),(56,113),(57,62),(58,115),(59,64),(60,117)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(15,30),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(56,60),(57,59),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,90),(82,89),(83,88),(84,87),(85,86),(91,94),(92,93),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 4 | 30 | 60 | 2 | 2 | 30 | 60 | 2 | 2 | 2 | 8 | 8 | 4 | 60 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D15 | D30 | D30 | C8⋊C22 | S3×D4 | D4×D5 | D8⋊S3 | D8⋊D5 | D4×D15 | D8⋊D15 |
| kernel | D8⋊D15 | C40⋊S3 | C24⋊D5 | D4⋊D15 | D4.D15 | C15×D8 | D4×D15 | D4⋊2D15 | C5×D8 | Dic15 | D30 | C3×D8 | C40 | C5×D4 | C24 | C3×D4 | D8 | C8 | D4 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D8⋊D15 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 239 | 0 |
| 0 | 0 | 0 | 1 | 0 | 239 |
| 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 94 | 53 | 147 | 188 |
| 0 | 0 | 188 | 147 | 53 | 94 |
| 0 | 0 | 47 | 147 | 147 | 188 |
| 0 | 0 | 94 | 194 | 53 | 94 |
| 52 | 240 | 0 | 0 | 0 | 0 |
| 53 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 2 | 239 |
| 0 | 0 | 240 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 51 | 0 | 0 | 0 | 0 |
| 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 2 | 0 |
| 0 | 0 | 240 | 1 | 2 | 239 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,239,0,240,0,0,0,0,239,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,94,188,47,94,0,0,53,147,147,194,0,0,147,53,147,53,0,0,188,94,188,94],[52,53,0,0,0,0,240,240,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,2,2,1,1,0,0,239,0,240,0],[0,52,0,0,0,0,51,0,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,2,2,1,1,0,0,0,239,0,240] >;
D8⋊D15 in GAP, Magma, Sage, TeX
D_8\rtimes D_{15} % in TeX
G:=Group("D8:D15"); // GroupNames label
G:=SmallGroup(480,876);
// by ID
G=gap.SmallGroup(480,876);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^30=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations