metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊3D10, D20.23D6, C60.4C23, D60.2C22, D4⋊D5⋊1S3, (C5×D4)⋊8D6, D4⋊2(S3×D5), C5⋊2C8⋊5D6, D4⋊D15⋊4C2, (C3×D4)⋊2D10, (S3×D20)⋊2C2, C5⋊7(D8⋊S3), D4⋊2S3⋊1D5, (C4×S3).5D10, (S3×C10).8D4, C3⋊2(D4⋊D10), C15⋊14(C8⋊C22), C15⋊3C8⋊4C22, C30.166(C2×D4), C30.D4⋊1C2, C10.140(S3×D4), Dic6⋊D5⋊1C2, D6.6(C5⋊D4), (D4×C15)⋊4C22, D6.Dic5⋊1C2, C20.4(C22×S3), C12.4(C22×D5), (S3×C20).2C22, (C5×Dic3).34D4, (C5×Dic6)⋊1C22, (C3×D20).1C22, Dic3.15(C5⋊D4), C4.4(C2×S3×D5), (C3×D4⋊D5)⋊2C2, C6.43(C2×C5⋊D4), C2.21(S3×C5⋊D4), (C5×D4⋊2S3)⋊1C2, (C3×C5⋊2C8)⋊2C22, SmallGroup(480,556)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60.C22
G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a19, dad=a31, cbc=a18b, dbd=a45b, dcd=a45c >
Subgroups: 876 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, Dic3, C12, D6, D6 [×3], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20, C20 [×2], D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C5⋊2C8, C5⋊2C8, D20, D20 [×2], C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C5×Dic3, C5×Dic3, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, D4⋊D5, Q8⋊D5 [×2], C2×D20, C5×C4○D4, D8⋊S3, C3×C5⋊2C8, C15⋊3C8, C3⋊D20, C3×D20, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D60, D4×C15, C2×S3×D5, D4⋊D10, D6.Dic5, C30.D4, Dic6⋊D5, C3×D4⋊D5, D4⋊D15, S3×D20, C5×D4⋊2S3, D60.C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D8⋊S3, C2×S3×D5, D4⋊D10, S3×C5⋊D4, D60.C22
►On 120 points
Generators in S120
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)
(2 20)(3 39)(4 58)(5 17)(6 36)(7 55)(8 14)(9 33)(10 52)(12 30)(13 49)(15 27)(16 46)(18 24)(19 43)(22 40)(23 59)(25 37)(26 56)(28 34)(29 53)(32 50)(35 47)(38 44)(42 60)(45 57)(48 54)(61 64)(62 83)(63 102)(65 80)(66 99)(67 118)(68 77)(69 96)(70 115)(71 74)(72 93)(73 112)(75 90)(76 109)(78 87)(79 106)(81 84)(82 103)(85 100)(86 119)(88 97)(89 116)(91 94)(92 113)(95 110)(98 107)(101 104)(105 120)(108 117)(111 114)
(1 70)(2 101)(3 72)(4 103)(5 74)(6 105)(7 76)(8 107)(9 78)(10 109)(11 80)(12 111)(13 82)(14 113)(15 84)(16 115)(17 86)(18 117)(19 88)(20 119)(21 90)(22 61)(23 92)(24 63)(25 94)(26 65)(27 96)(28 67)(29 98)(30 69)(31 100)(32 71)(33 102)(34 73)(35 104)(36 75)(37 106)(38 77)(39 108)(40 79)(41 110)(42 81)(43 112)(44 83)(45 114)(46 85)(47 116)(48 87)(49 118)(50 89)(51 120)(52 91)(53 62)(54 93)(55 64)(56 95)(57 66)(58 97)(59 68)(60 99)
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108), (2,20)(3,39)(4,58)(5,17)(6,36)(7,55)(8,14)(9,33)(10,52)(12,30)(13,49)(15,27)(16,46)(18,24)(19,43)(22,40)(23,59)(25,37)(26,56)(28,34)(29,53)(32,50)(35,47)(38,44)(42,60)(45,57)(48,54)(61,64)(62,83)(63,102)(65,80)(66,99)(67,118)(68,77)(69,96)(70,115)(71,74)(72,93)(73,112)(75,90)(76,109)(78,87)(79,106)(81,84)(82,103)(85,100)(86,119)(88,97)(89,116)(91,94)(92,113)(95,110)(98,107)(101,104)(105,120)(108,117)(111,114), (1,70)(2,101)(3,72)(4,103)(5,74)(6,105)(7,76)(8,107)(9,78)(10,109)(11,80)(12,111)(13,82)(14,113)(15,84)(16,115)(17,86)(18,117)(19,88)(20,119)(21,90)(22,61)(23,92)(24,63)(25,94)(26,65)(27,96)(28,67)(29,98)(30,69)(31,100)(32,71)(33,102)(34,73)(35,104)(36,75)(37,106)(38,77)(39,108)(40,79)(41,110)(42,81)(43,112)(44,83)(45,114)(46,85)(47,116)(48,87)(49,118)(50,89)(51,120)(52,91)(53,62)(54,93)(55,64)(56,95)(57,66)(58,97)(59,68)(60,99)>;
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108), (2,20)(3,39)(4,58)(5,17)(6,36)(7,55)(8,14)(9,33)(10,52)(12,30)(13,49)(15,27)(16,46)(18,24)(19,43)(22,40)(23,59)(25,37)(26,56)(28,34)(29,53)(32,50)(35,47)(38,44)(42,60)(45,57)(48,54)(61,64)(62,83)(63,102)(65,80)(66,99)(67,118)(68,77)(69,96)(70,115)(71,74)(72,93)(73,112)(75,90)(76,109)(78,87)(79,106)(81,84)(82,103)(85,100)(86,119)(88,97)(89,116)(91,94)(92,113)(95,110)(98,107)(101,104)(105,120)(108,117)(111,114), (1,70)(2,101)(3,72)(4,103)(5,74)(6,105)(7,76)(8,107)(9,78)(10,109)(11,80)(12,111)(13,82)(14,113)(15,84)(16,115)(17,86)(18,117)(19,88)(20,119)(21,90)(22,61)(23,92)(24,63)(25,94)(26,65)(27,96)(28,67)(29,98)(30,69)(31,100)(32,71)(33,102)(34,73)(35,104)(36,75)(37,106)(38,77)(39,108)(40,79)(41,110)(42,81)(43,112)(44,83)(45,114)(46,85)(47,116)(48,87)(49,118)(50,89)(51,120)(52,91)(53,62)(54,93)(55,64)(56,95)(57,66)(58,97)(59,68)(60,99) );
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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82),(73,81),(74,80),(75,79),(76,78),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108)], [(2,20),(3,39),(4,58),(5,17),(6,36),(7,55),(8,14),(9,33),(10,52),(12,30),(13,49),(15,27),(16,46),(18,24),(19,43),(22,40),(23,59),(25,37),(26,56),(28,34),(29,53),(32,50),(35,47),(38,44),(42,60),(45,57),(48,54),(61,64),(62,83),(63,102),(65,80),(66,99),(67,118),(68,77),(69,96),(70,115),(71,74),(72,93),(73,112),(75,90),(76,109),(78,87),(79,106),(81,84),(82,103),(85,100),(86,119),(88,97),(89,116),(91,94),(92,113),(95,110),(98,107),(101,104),(105,120),(108,117),(111,114)], [(1,70),(2,101),(3,72),(4,103),(5,74),(6,105),(7,76),(8,107),(9,78),(10,109),(11,80),(12,111),(13,82),(14,113),(15,84),(16,115),(17,86),(18,117),(19,88),(20,119),(21,90),(22,61),(23,92),(24,63),(25,94),(26,65),(27,96),(28,67),(29,98),(30,69),(31,100),(32,71),(33,102),(34,73),(35,104),(36,75),(37,106),(38,77),(39,108),(40,79),(41,110),(42,81),(43,112),(44,83),(45,114),(46,85),(47,116),(48,87),(49,118),(50,89),(51,120),(52,91),(53,62),(54,93),(55,64),(56,95),(57,66),(58,97),(59,68),(60,99)])
48 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 | 10G | 10H | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 6 | 20 | 60 | 2 | 2 | 6 | 12 | 2 | 2 | 2 | 8 | 40 | 20 | 60 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D4 | S3×D5 | D8⋊S3 | C2×S3×D5 | D4⋊D10 | S3×C5⋊D4 | D60.C22 |
| kernel | D60.C22 | D6.Dic5 | C30.D4 | Dic6⋊D5 | C3×D4⋊D5 | D4⋊D15 | S3×D20 | C5×D4⋊2S3 | D4⋊D5 | C5×Dic3 | S3×C10 | D4⋊2S3 | C5⋊2C8 | D20 | C5×D4 | Dic6 | C4×S3 | C3×D4 | Dic3 | D6 | C15 | C10 | D4 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D60.C22 ►in GL6(𝔽241)
| 189 | 51 | 0 | 0 | 0 | 0 |
| 189 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 189 | 1 | 0 | 0 | 0 | 0 |
| 189 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 94 | 47 | 94 |
| 0 | 0 | 147 | 194 | 147 | 194 |
| 0 | 0 | 47 | 94 | 194 | 147 |
| 0 | 0 | 147 | 194 | 94 | 47 |
G:=sub<GL(6,GF(241))| [189,189,0,0,0,0,51,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,240,1,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,240,1,0,0,240,0,0,0,0,0,240,1,0,0],[189,189,0,0,0,0,1,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,94,194,94,194,0,0,47,147,194,94,0,0,94,194,147,47] >;
D60.C22 in GAP, Magma, Sage, TeX
D_{60}.C_2^2 % in TeX
G:=Group("D60.C2^2"); // GroupNames label
G:=SmallGroup(480,556);
// by ID
G=gap.SmallGroup(480,556);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^60=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^19,d*a*d=a^31,c*b*c=a^18*b,d*b*d=a^45*b,d*c*d=a^45*c>;
// generators/relations