metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D15, C4.5D30, C20.19D6, Dic30⋊3C2, C12.19D10, C60.5C22, C22.1D30, C30.33C23, D30.6C22, Dic15.16C22, (C3×D4)⋊3D5, (C5×D4)⋊3S3, (D4×C15)⋊3C2, (C4×D15)⋊2C2, C15⋊7D4⋊2C2, (C2×C10).3D6, (C2×C6).3D10, C15⋊14(C4○D4), C3⋊5(D4⋊2D5), C5⋊5(D4⋊2S3), (C2×Dic15)⋊3C2, (C2×C30).1C22, C6.33(C22×D5), C2.7(C22×D15), C10.33(C22×S3), SmallGroup(240,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D15
G = < a,b,c,d | a4=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 352 in 80 conjugacy classes, 35 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, D15, C30, C30, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, D4⋊2S3, Dic15, Dic15, C60, D30, C2×C30, D4⋊2D5, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, D4⋊2D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, D15, C22×D5, D4⋊2S3, D30, D4⋊2D5, C22×D15, D4⋊2D15
►On 120 points
Generators in S120
(1 84 17 73)(2 85 18 74)(3 86 19 75)(4 87 20 61)(5 88 21 62)(6 89 22 63)(7 90 23 64)(8 76 24 65)(9 77 25 66)(10 78 26 67)(11 79 27 68)(12 80 28 69)(13 81 29 70)(14 82 30 71)(15 83 16 72)(31 113 46 91)(32 114 47 92)(33 115 48 93)(34 116 49 94)(35 117 50 95)(36 118 51 96)(37 119 52 97)(38 120 53 98)(39 106 54 99)(40 107 55 100)(41 108 56 101)(42 109 57 102)(43 110 58 103)(44 111 59 104)(45 112 60 105)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 91)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 72)(32 73)(33 74)(34 75)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)
(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 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 47)(32 46)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)
G:=sub<Sym(120)| (1,84,17,73)(2,85,18,74)(3,86,19,75)(4,87,20,61)(5,88,21,62)(6,89,22,63)(7,90,23,64)(8,76,24,65)(9,77,25,66)(10,78,26,67)(11,79,27,68)(12,80,28,69)(13,81,29,70)(14,82,30,71)(15,83,16,72)(31,113,46,91)(32,114,47,92)(33,115,48,93)(34,116,49,94)(35,117,50,95)(36,118,51,96)(37,119,52,97)(38,120,53,98)(39,106,54,99)(40,107,55,100)(41,108,56,101)(42,109,57,102)(43,110,58,103)(44,111,59,104)(45,112,60,105), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,91)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,72)(32,73)(33,74)(34,75)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82), (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)(16,17)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(31,47)(32,46)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)>;
G:=Group( (1,84,17,73)(2,85,18,74)(3,86,19,75)(4,87,20,61)(5,88,21,62)(6,89,22,63)(7,90,23,64)(8,76,24,65)(9,77,25,66)(10,78,26,67)(11,79,27,68)(12,80,28,69)(13,81,29,70)(14,82,30,71)(15,83,16,72)(31,113,46,91)(32,114,47,92)(33,115,48,93)(34,116,49,94)(35,117,50,95)(36,118,51,96)(37,119,52,97)(38,120,53,98)(39,106,54,99)(40,107,55,100)(41,108,56,101)(42,109,57,102)(43,110,58,103)(44,111,59,104)(45,112,60,105), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,91)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,72)(32,73)(33,74)(34,75)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82), (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)(16,17)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(31,47)(32,46)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115) );
G=PermutationGroup([[(1,84,17,73),(2,85,18,74),(3,86,19,75),(4,87,20,61),(5,88,21,62),(6,89,22,63),(7,90,23,64),(8,76,24,65),(9,77,25,66),(10,78,26,67),(11,79,27,68),(12,80,28,69),(13,81,29,70),(14,82,30,71),(15,83,16,72),(31,113,46,91),(32,114,47,92),(33,115,48,93),(34,116,49,94),(35,117,50,95),(36,118,51,96),(37,119,52,97),(38,120,53,98),(39,106,54,99),(40,107,55,100),(41,108,56,101),(42,109,57,102),(43,110,58,103),(44,111,59,104),(45,112,60,105)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,100),(10,101),(11,102),(12,103),(13,104),(14,105),(15,91),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,72),(32,73),(33,74),(34,75),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,67),(42,68),(43,69),(44,70),(45,71),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82)], [(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),(16,17),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(31,47),(32,46),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(61,69),(62,68),(63,67),(64,66),(70,75),(71,74),(72,73),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(99,106),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115)]])
D4⋊2D15 is a maximal subgroup of
D30.8D4 D30.9D4 D20.10D6 D30.11D4 D8⋊D15 D8⋊3D15 SD16⋊D15 D4.5D30 C15⋊2- 1+4 D5×D4⋊2S3 S3×D4⋊2D5 D20⋊13D6 D4⋊6D30 C4○D4×D15 D4.10D30
D4⋊2D15 is a maximal quotient of
C23.15D30 C22⋊2Dic30 C23.8D30 Dic15⋊19D4 D30.28D4 C23.11D30 C22.D60 Dic15⋊10Q8 Dic15.3Q8 C4.Dic30 C4⋊C4⋊7D15 D30⋊6Q8 C4⋊C4⋊D15 D4×Dic15 C23.22D30 C60.17D4 C60⋊2D4 Dic15⋊12D4
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 30 | 2 | 2 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D15 | D30 | D30 | D4⋊2S3 | D4⋊2D5 | D4⋊2D15 |
| kernel | D4⋊2D15 | Dic30 | C4×D15 | C2×Dic15 | C15⋊7D4 | D4×C15 | C5×D4 | C3×D4 | C20 | C2×C10 | C15 | C12 | C2×C6 | D4 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 4 |
Matrix representation of D4⋊2D15 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 56 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 44 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,50,0],[27,56,0,0,0,0,5,33,0,0,0,0,0,0,44,44,0,0,0,0,17,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;
D4⋊2D15 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{15} % in TeX
G:=Group("D4:2D15"); // GroupNames label
G:=SmallGroup(240,180);
// by ID
G=gap.SmallGroup(240,180);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations