direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8×D15, C8⋊4D30, C40⋊6D6, D4⋊1D30, C24⋊6D10, D120⋊7C2, C120⋊3C22, D30.48D4, D60⋊14C22, C60.64C23, Dic15.22D4, C5⋊4(S3×D8), C3⋊4(D5×D8), (C3×D8)⋊2D5, (C5×D4)⋊5D6, (C5×D8)⋊2S3, C15⋊13(C2×D8), D4⋊D15⋊9C2, (C3×D4)⋊5D10, (C15×D8)⋊2C2, (D4×D15)⋊8C2, (C8×D15)⋊1C2, C6.108(D4×D5), C2.15(D4×D15), C10.110(S3×D4), C30.315(C2×D4), C4.1(C22×D15), C15⋊3C8⋊27C22, (D4×C15)⋊14C22, C20.102(C22×S3), (C4×D15).42C22, C12.102(C22×D5), SmallGroup(480,875)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1300 in 152 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, D8, C2×D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, D15, D15, C30, C30, C2×D8, C5⋊2C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, Dic15, C60, D30, D30, C2×C30, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, S3×D8, C15⋊3C8, C120, C4×D15, D60, C15⋊7D4, D4×C15, C22×D15, D5×D8, C8×D15, D120, D4⋊D15, C15×D8, D4×D15, D8×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, D15, C2×D8, C22×D5, S3×D4, D30, D4×D5, S3×D8, C22×D15, D5×D8, D4×D15, D8×D15
►On 120 points
Generators in S120
(1 95 39 61 20 106 54 84)(2 96 40 62 21 107 55 85)(3 97 41 63 22 108 56 86)(4 98 42 64 23 109 57 87)(5 99 43 65 24 110 58 88)(6 100 44 66 25 111 59 89)(7 101 45 67 26 112 60 90)(8 102 31 68 27 113 46 76)(9 103 32 69 28 114 47 77)(10 104 33 70 29 115 48 78)(11 105 34 71 30 116 49 79)(12 91 35 72 16 117 50 80)(13 92 36 73 17 118 51 81)(14 93 37 74 18 119 52 82)(15 94 38 75 19 120 53 83)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 76)(28 77)(29 78)(30 79)(31 102)(32 103)(33 104)(34 105)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)
(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 19)(2 18)(3 17)(4 16)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(91 109)(92 108)(93 107)(94 106)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)
G:=sub<Sym(120)| (1,95,39,61,20,106,54,84)(2,96,40,62,21,107,55,85)(3,97,41,63,22,108,56,86)(4,98,42,64,23,109,57,87)(5,99,43,65,24,110,58,88)(6,100,44,66,25,111,59,89)(7,101,45,67,26,112,60,90)(8,102,31,68,27,113,46,76)(9,103,32,69,28,114,47,77)(10,104,33,70,29,115,48,78)(11,105,34,71,30,116,49,79)(12,91,35,72,16,117,50,80)(13,92,36,73,17,118,51,81)(14,93,37,74,18,119,52,82)(15,94,38,75,19,120,53,83), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,76)(28,77)(29,78)(30,79)(31,102)(32,103)(33,104)(34,105)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112), (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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(91,109)(92,108)(93,107)(94,106)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)>;
G:=Group( (1,95,39,61,20,106,54,84)(2,96,40,62,21,107,55,85)(3,97,41,63,22,108,56,86)(4,98,42,64,23,109,57,87)(5,99,43,65,24,110,58,88)(6,100,44,66,25,111,59,89)(7,101,45,67,26,112,60,90)(8,102,31,68,27,113,46,76)(9,103,32,69,28,114,47,77)(10,104,33,70,29,115,48,78)(11,105,34,71,30,116,49,79)(12,91,35,72,16,117,50,80)(13,92,36,73,17,118,51,81)(14,93,37,74,18,119,52,82)(15,94,38,75,19,120,53,83), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,76)(28,77)(29,78)(30,79)(31,102)(32,103)(33,104)(34,105)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112), (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), (1,19)(2,18)(3,17)(4,16)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(91,109)(92,108)(93,107)(94,106)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110) );
G=PermutationGroup([[(1,95,39,61,20,106,54,84),(2,96,40,62,21,107,55,85),(3,97,41,63,22,108,56,86),(4,98,42,64,23,109,57,87),(5,99,43,65,24,110,58,88),(6,100,44,66,25,111,59,89),(7,101,45,67,26,112,60,90),(8,102,31,68,27,113,46,76),(9,103,32,69,28,114,47,77),(10,104,33,70,29,115,48,78),(11,105,34,71,30,116,49,79),(12,91,35,72,16,117,50,80),(13,92,36,73,17,118,51,81),(14,93,37,74,18,119,52,82),(15,94,38,75,19,120,53,83)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,76),(28,77),(29,78),(30,79),(31,102),(32,103),(33,104),(34,105),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(31,46),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(91,109),(92,108),(93,107),(94,106),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110)]])
63 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 | 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 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 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 | 15 | 15 | 60 | 60 | 2 | 2 | 30 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 30 | 30 | 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 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | D15 | D30 | D30 | S3×D4 | D4×D5 | S3×D8 | D5×D8 | D4×D15 | D8×D15 |
| kernel | D8×D15 | C8×D15 | D120 | D4⋊D15 | C15×D8 | D4×D15 | C5×D8 | Dic15 | D30 | C3×D8 | C40 | C5×D4 | D15 | C24 | C3×D4 | D8 | C8 | D4 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D8×D15 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 208 |
| 0 | 0 | 168 | 219 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 208 |
| 0 | 0 | 168 | 219 |
| 16 | 68 | 0 | 0 |
| 173 | 178 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 64 | 0 | 0 |
| 178 | 211 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,168,0,0,208,219],[1,0,0,0,0,1,0,0,0,0,22,168,0,0,208,219],[16,173,0,0,68,178,0,0,0,0,1,0,0,0,0,1],[30,178,0,0,64,211,0,0,0,0,240,0,0,0,0,240] >;
D8×D15 in GAP, Magma, Sage, TeX
D_8\times D_{15} % in TeX
G:=Group("D8xD15"); // GroupNames label
G:=SmallGroup(480,875);
// by ID
G=gap.SmallGroup(480,875);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,346,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations