metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊24D6, D12⋊24D10, Dic6⋊22D10, Dic10⋊22D6, C30.20C24, C60.163C23, D30.38C23, Dic15.38C23, (C2×C20)⋊7D6, C5⋊D4⋊8D6, C4○D20⋊7S3, C4○D12⋊7D5, (C4×D5)⋊13D6, (C2×C12)⋊7D10, C3⋊D4⋊8D10, (C4×S3)⋊13D10, D15⋊Q8⋊13C2, C15⋊Q8⋊11C22, D15⋊1(C4○D4), D10⋊D6⋊7C2, C20⋊D6⋊13C2, (C2×C60)⋊17C22, D20⋊S3⋊13C2, D12⋊D5⋊13C2, D6.6(C22×D5), (C6×D5).6C23, C6.20(C23×D5), (S3×C20)⋊13C22, C30.C23⋊7C2, (C3×D20)⋊31C22, (C5×D12)⋊31C22, (D5×C12)⋊13C22, (C4×D15)⋊24C22, C3⋊D20⋊13C22, C15⋊D4⋊13C22, C5⋊D12⋊13C22, (S3×C10).6C23, C10.20(S3×C23), (D5×Dic3)⋊8C22, (S3×Dic5)⋊8C22, D10.6(C22×S3), D6.D10⋊11C2, (C2×C30).239C23, C20.189(C22×S3), (C5×Dic6)⋊28C22, C12.189(C22×D5), (C3×Dic5).9C23, Dic3.9(C22×D5), (C5×Dic3).9C23, Dic5.9(C22×S3), (C3×Dic10)⋊28C22, (C2×Dic15)⋊34C22, D30.C2.10C22, (C22×D15).122C22, C3⋊2(D5×C4○D4), C5⋊2(S3×C4○D4), (C4×S3×D5)⋊10C2, (C2×C4×D15)⋊27C2, (C2×C4)⋊15(S3×D5), C15⋊10(C2×C4○D4), C4.162(C2×S3×D5), (C3×C4○D20)⋊10C2, (C5×C4○D12)⋊10C2, (C2×S3×D5).7C22, C2.23(C22×S3×D5), C22.18(C2×S3×D5), (C5×C3⋊D4)⋊9C22, (C3×C5⋊D4)⋊9C22, (C2×C6).11(C22×D5), (C2×C10).11(C22×S3), SmallGroup(480,1092)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊24D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >
Subgroups: 1676 in 328 conjugacy classes, 110 normal (52 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, D15, C30, C30, C2×C4○D4, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, S3×C2×C4, C4○D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D30, C2×C30, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C4×D15, C2×Dic15, C2×C60, C2×S3×D5, C22×D15, D5×C4○D4, D20⋊S3, D12⋊D5, D15⋊Q8, D6.D10, C4×S3×D5, C20⋊D6, C30.C23, D10⋊D6, C3×C4○D20, C5×C4○D12, C2×C4×D15, D20⋊24D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5, D5×C4○D4, C22×S3×D5, D20⋊24D6
►On 120 points
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 120 85 40 64 55)(2 101 86 21 65 56)(3 102 87 22 66 57)(4 103 88 23 67 58)(5 104 89 24 68 59)(6 105 90 25 69 60)(7 106 91 26 70 41)(8 107 92 27 71 42)(9 108 93 28 72 43)(10 109 94 29 73 44)(11 110 95 30 74 45)(12 111 96 31 75 46)(13 112 97 32 76 47)(14 113 98 33 77 48)(15 114 99 34 78 49)(16 115 100 35 79 50)(17 116 81 36 80 51)(18 117 82 37 61 52)(19 118 83 38 62 53)(20 119 84 39 63 54)
(1 45)(2 54)(3 43)(4 52)(5 41)(6 50)(7 59)(8 48)(9 57)(10 46)(11 55)(12 44)(13 53)(14 42)(15 51)(16 60)(17 49)(18 58)(19 47)(20 56)(21 84)(22 93)(23 82)(24 91)(25 100)(26 89)(27 98)(28 87)(29 96)(30 85)(31 94)(32 83)(33 92)(34 81)(35 90)(36 99)(37 88)(38 97)(39 86)(40 95)(61 103)(62 112)(63 101)(64 110)(65 119)(66 108)(67 117)(68 106)(69 115)(70 104)(71 113)(72 102)(73 111)(74 120)(75 109)(76 118)(77 107)(78 116)(79 105)(80 114)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,88)(82,87)(83,86)(84,85)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115), (1,120,85,40,64,55)(2,101,86,21,65,56)(3,102,87,22,66,57)(4,103,88,23,67,58)(5,104,89,24,68,59)(6,105,90,25,69,60)(7,106,91,26,70,41)(8,107,92,27,71,42)(9,108,93,28,72,43)(10,109,94,29,73,44)(11,110,95,30,74,45)(12,111,96,31,75,46)(13,112,97,32,76,47)(14,113,98,33,77,48)(15,114,99,34,78,49)(16,115,100,35,79,50)(17,116,81,36,80,51)(18,117,82,37,61,52)(19,118,83,38,62,53)(20,119,84,39,63,54), (1,45)(2,54)(3,43)(4,52)(5,41)(6,50)(7,59)(8,48)(9,57)(10,46)(11,55)(12,44)(13,53)(14,42)(15,51)(16,60)(17,49)(18,58)(19,47)(20,56)(21,84)(22,93)(23,82)(24,91)(25,100)(26,89)(27,98)(28,87)(29,96)(30,85)(31,94)(32,83)(33,92)(34,81)(35,90)(36,99)(37,88)(38,97)(39,86)(40,95)(61,103)(62,112)(63,101)(64,110)(65,119)(66,108)(67,117)(68,106)(69,115)(70,104)(71,113)(72,102)(73,111)(74,120)(75,109)(76,118)(77,107)(78,116)(79,105)(80,114)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,88)(82,87)(83,86)(84,85)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115), (1,120,85,40,64,55)(2,101,86,21,65,56)(3,102,87,22,66,57)(4,103,88,23,67,58)(5,104,89,24,68,59)(6,105,90,25,69,60)(7,106,91,26,70,41)(8,107,92,27,71,42)(9,108,93,28,72,43)(10,109,94,29,73,44)(11,110,95,30,74,45)(12,111,96,31,75,46)(13,112,97,32,76,47)(14,113,98,33,77,48)(15,114,99,34,78,49)(16,115,100,35,79,50)(17,116,81,36,80,51)(18,117,82,37,61,52)(19,118,83,38,62,53)(20,119,84,39,63,54), (1,45)(2,54)(3,43)(4,52)(5,41)(6,50)(7,59)(8,48)(9,57)(10,46)(11,55)(12,44)(13,53)(14,42)(15,51)(16,60)(17,49)(18,58)(19,47)(20,56)(21,84)(22,93)(23,82)(24,91)(25,100)(26,89)(27,98)(28,87)(29,96)(30,85)(31,94)(32,83)(33,92)(34,81)(35,90)(36,99)(37,88)(38,97)(39,86)(40,95)(61,103)(62,112)(63,101)(64,110)(65,119)(66,108)(67,117)(68,106)(69,115)(70,104)(71,113)(72,102)(73,111)(74,120)(75,109)(76,118)(77,107)(78,116)(79,105)(80,114) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,88),(82,87),(83,86),(84,85),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(101,108),(102,107),(103,106),(104,105),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115)], [(1,120,85,40,64,55),(2,101,86,21,65,56),(3,102,87,22,66,57),(4,103,88,23,67,58),(5,104,89,24,68,59),(6,105,90,25,69,60),(7,106,91,26,70,41),(8,107,92,27,71,42),(9,108,93,28,72,43),(10,109,94,29,73,44),(11,110,95,30,74,45),(12,111,96,31,75,46),(13,112,97,32,76,47),(14,113,98,33,77,48),(15,114,99,34,78,49),(16,115,100,35,79,50),(17,116,81,36,80,51),(18,117,82,37,61,52),(19,118,83,38,62,53),(20,119,84,39,63,54)], [(1,45),(2,54),(3,43),(4,52),(5,41),(6,50),(7,59),(8,48),(9,57),(10,46),(11,55),(12,44),(13,53),(14,42),(15,51),(16,60),(17,49),(18,58),(19,47),(20,56),(21,84),(22,93),(23,82),(24,91),(25,100),(26,89),(27,98),(28,87),(29,96),(30,85),(31,94),(32,83),(33,92),(34,81),(35,90),(36,99),(37,88),(38,97),(39,86),(40,95),(61,103),(62,112),(63,101),(64,110),(65,119),(66,108),(67,117),(68,106),(69,115),(70,104),(71,113),(72,102),(73,111),(74,120),(75,109),(76,118),(77,107),(78,116),(79,105),(80,114)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 30 | 2 | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | D10 | S3×D5 | S3×C4○D4 | C2×S3×D5 | C2×S3×D5 | D5×C4○D4 | D20⋊24D6 |
| kernel | D20⋊24D6 | D20⋊S3 | D12⋊D5 | D15⋊Q8 | D6.D10 | C4×S3×D5 | C20⋊D6 | C30.C23 | D10⋊D6 | C3×C4○D20 | C5×C4○D12 | C2×C4×D15 | C4○D20 | C4○D12 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | D15 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of D20⋊24D6 ►in GL6(𝔽61)
| 44 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 44 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 41 |
| 0 | 0 | 0 | 0 | 52 | 59 |
| 44 | 17 | 0 | 0 | 0 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 41 |
| 0 | 0 | 0 | 0 | 52 | 59 |
G:=sub<GL(6,GF(61))| [44,1,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,44,1,0,0,0,0,0,0,0,11,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,52,0,0,0,0,41,59],[44,1,0,0,0,0,17,17,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,2,52,0,0,0,0,41,59] >;
D20⋊24D6 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{24}D_6 % in TeX
G:=Group("D20:24D6"); // GroupNames label
G:=SmallGroup(480,1092);
// by ID
G=gap.SmallGroup(480,1092);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations