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○D12⋊7D5, C4○D20⋊7S3, (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, D12⋊D5⋊13C2, D20⋊S3⋊13C2, (C6×D5).6C23, D6.6(C22×D5), C6.20(C23×D5), (S3×C20)⋊13C22, C30.C23⋊7C2, (C4×D15)⋊24C22, (D5×C12)⋊13C22, (C5×D12)⋊31C22, (C3×D20)⋊31C22, C3⋊D20⋊13C22, C15⋊D4⋊13C22, C5⋊D12⋊13C22, (S3×C10).6C23, C10.20(S3×C23), (S3×Dic5)⋊8C22, (D5×Dic3)⋊8C22, D10.6(C22×S3), D6.D10⋊11C2, (C2×C30).239C23, C20.189(C22×S3), (C5×Dic6)⋊28C22, C12.189(C22×D5), (C5×Dic3).9C23, Dic5.9(C22×S3), (C3×Dic5).9C23, Dic3.9(C22×D5), (C3×Dic10)⋊28C22, (C2×Dic15)⋊34C22, D30.C2.10C22, (C22×D15).122C22, C5⋊2(S3×C4○D4), C3⋊2(D5×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
Subgroups: 1676 in 328 conjugacy classes, 110 normal (52 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×5], C6, C6 [×3], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×5], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6 [×2], C4×S3 [×2], C4×S3 [×8], D12, D12 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×3], C5×S3 [×2], C3×D5 [×2], D15 [×2], D15, C30, C30, C2×C4○D4, Dic10, Dic10 [×2], C4×D5 [×2], C4×D5 [×8], D20, D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12, C4○D12 [×2], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C4×D5 [×3], C4○D20, C4○D20 [×2], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C2×S3×D5 [×2], C22×D15, D5×C4○D4, D20⋊S3, D12⋊D5, D15⋊Q8, D6.D10 [×2], C4×S3×D5 [×2], C20⋊D6, C30.C23 [×2], D10⋊D6 [×2], C3×C4○D20, C5×C4○D12, C2×C4×D15, D20⋊24D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D20⋊24D6
Generators and relations
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 >
►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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 86 24 77 112 51)(2 87 25 78 113 52)(3 88 26 79 114 53)(4 89 27 80 115 54)(5 90 28 61 116 55)(6 91 29 62 117 56)(7 92 30 63 118 57)(8 93 31 64 119 58)(9 94 32 65 120 59)(10 95 33 66 101 60)(11 96 34 67 102 41)(12 97 35 68 103 42)(13 98 36 69 104 43)(14 99 37 70 105 44)(15 100 38 71 106 45)(16 81 39 72 107 46)(17 82 40 73 108 47)(18 83 21 74 109 48)(19 84 22 75 110 49)(20 85 23 76 111 50)
(1 41)(2 50)(3 59)(4 48)(5 57)(6 46)(7 55)(8 44)(9 53)(10 42)(11 51)(12 60)(13 49)(14 58)(15 47)(16 56)(17 45)(18 54)(19 43)(20 52)(21 80)(22 69)(23 78)(24 67)(25 76)(26 65)(27 74)(28 63)(29 72)(30 61)(31 70)(32 79)(33 68)(34 77)(35 66)(36 75)(37 64)(38 73)(39 62)(40 71)(81 117)(82 106)(83 115)(84 104)(85 113)(86 102)(87 111)(88 120)(89 109)(90 118)(91 107)(92 116)(93 105)(94 114)(95 103)(96 112)(97 101)(98 110)(99 119)(100 108)
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,86,24,77,112,51)(2,87,25,78,113,52)(3,88,26,79,114,53)(4,89,27,80,115,54)(5,90,28,61,116,55)(6,91,29,62,117,56)(7,92,30,63,118,57)(8,93,31,64,119,58)(9,94,32,65,120,59)(10,95,33,66,101,60)(11,96,34,67,102,41)(12,97,35,68,103,42)(13,98,36,69,104,43)(14,99,37,70,105,44)(15,100,38,71,106,45)(16,81,39,72,107,46)(17,82,40,73,108,47)(18,83,21,74,109,48)(19,84,22,75,110,49)(20,85,23,76,111,50), (1,41)(2,50)(3,59)(4,48)(5,57)(6,46)(7,55)(8,44)(9,53)(10,42)(11,51)(12,60)(13,49)(14,58)(15,47)(16,56)(17,45)(18,54)(19,43)(20,52)(21,80)(22,69)(23,78)(24,67)(25,76)(26,65)(27,74)(28,63)(29,72)(30,61)(31,70)(32,79)(33,68)(34,77)(35,66)(36,75)(37,64)(38,73)(39,62)(40,71)(81,117)(82,106)(83,115)(84,104)(85,113)(86,102)(87,111)(88,120)(89,109)(90,118)(91,107)(92,116)(93,105)(94,114)(95,103)(96,112)(97,101)(98,110)(99,119)(100,108)>;
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,86,24,77,112,51)(2,87,25,78,113,52)(3,88,26,79,114,53)(4,89,27,80,115,54)(5,90,28,61,116,55)(6,91,29,62,117,56)(7,92,30,63,118,57)(8,93,31,64,119,58)(9,94,32,65,120,59)(10,95,33,66,101,60)(11,96,34,67,102,41)(12,97,35,68,103,42)(13,98,36,69,104,43)(14,99,37,70,105,44)(15,100,38,71,106,45)(16,81,39,72,107,46)(17,82,40,73,108,47)(18,83,21,74,109,48)(19,84,22,75,110,49)(20,85,23,76,111,50), (1,41)(2,50)(3,59)(4,48)(5,57)(6,46)(7,55)(8,44)(9,53)(10,42)(11,51)(12,60)(13,49)(14,58)(15,47)(16,56)(17,45)(18,54)(19,43)(20,52)(21,80)(22,69)(23,78)(24,67)(25,76)(26,65)(27,74)(28,63)(29,72)(30,61)(31,70)(32,79)(33,68)(34,77)(35,66)(36,75)(37,64)(38,73)(39,62)(40,71)(81,117)(82,106)(83,115)(84,104)(85,113)(86,102)(87,111)(88,120)(89,109)(90,118)(91,107)(92,116)(93,105)(94,114)(95,103)(96,112)(97,101)(98,110)(99,119)(100,108) );
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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,86,24,77,112,51),(2,87,25,78,113,52),(3,88,26,79,114,53),(4,89,27,80,115,54),(5,90,28,61,116,55),(6,91,29,62,117,56),(7,92,30,63,118,57),(8,93,31,64,119,58),(9,94,32,65,120,59),(10,95,33,66,101,60),(11,96,34,67,102,41),(12,97,35,68,103,42),(13,98,36,69,104,43),(14,99,37,70,105,44),(15,100,38,71,106,45),(16,81,39,72,107,46),(17,82,40,73,108,47),(18,83,21,74,109,48),(19,84,22,75,110,49),(20,85,23,76,111,50)], [(1,41),(2,50),(3,59),(4,48),(5,57),(6,46),(7,55),(8,44),(9,53),(10,42),(11,51),(12,60),(13,49),(14,58),(15,47),(16,56),(17,45),(18,54),(19,43),(20,52),(21,80),(22,69),(23,78),(24,67),(25,76),(26,65),(27,74),(28,63),(29,72),(30,61),(31,70),(32,79),(33,68),(34,77),(35,66),(36,75),(37,64),(38,73),(39,62),(40,71),(81,117),(82,106),(83,115),(84,104),(85,113),(86,102),(87,111),(88,120),(89,109),(90,118),(91,107),(92,116),(93,105),(94,114),(95,103),(96,112),(97,101),(98,110),(99,119),(100,108)])
Matrix representation ►G ⊆ 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] >;
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 |
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