metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.12D4, D20.12D6, D60⋊6C22, D12.12D10, C60.30C23, Dic15.44D4, C3⋊C8⋊12D10, Q8⋊D5⋊3S3, (C5×Q8)⋊7D6, Q8⋊7(S3×D5), C5⋊D24⋊6C2, C5⋊2C8⋊12D6, C3⋊D40⋊6C2, C5⋊4(Q8⋊3D6), (C3×Q8)⋊4D10, C6.76(D4×D5), C20⋊D6⋊3C2, C3⋊4(D40⋊C2), Q8⋊2S3⋊3D5, C10.77(S3×D4), Q8⋊3D15⋊1C2, C15⋊24(C8⋊C22), C30.192(C2×D4), (Q8×C15)⋊6C22, D30.5C4⋊5C2, C20.30(C22×S3), C12.30(C22×D5), (C3×D20).11C22, (C4×D15).10C22, (C5×D12).11C22, C2.29(D10⋊D6), C4.30(C2×S3×D5), (C3×Q8⋊D5)⋊4C2, (C5×C3⋊C8)⋊14C22, (C5×Q8⋊2S3)⋊4C2, (C3×C5⋊2C8)⋊14C22, SmallGroup(480,582)
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=a11, dad=a29, cbc=a55b, dbd=a58b, cd=dc >
Subgroups: 1020 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, Dic15, C60, C60, S3×D5, C6×D5, S3×C10, D30, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, Q8⋊3D6, C5×C3⋊C8, C3×C5⋊2C8, C15⋊D4, C3×D20, C5×D12, C4×D15, C4×D15, D60, D60, Q8×C15, C2×S3×D5, D40⋊C2, D30.5C4, C3⋊D40, C5⋊D24, C3×Q8⋊D5, C5×Q8⋊2S3, C20⋊D6, Q8⋊3D15, D60⋊C22
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q8⋊3D6, C2×S3×D5, D40⋊C2, D10⋊D6, 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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 120)(20 119)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)
(2 12)(3 23)(4 34)(5 45)(6 56)(8 18)(9 29)(10 40)(11 51)(14 24)(15 35)(16 46)(17 57)(20 30)(21 41)(22 52)(26 36)(27 47)(28 58)(32 42)(33 53)(38 48)(39 59)(44 54)(50 60)(61 76)(62 87)(63 98)(64 109)(65 120)(66 71)(67 82)(68 93)(69 104)(70 115)(72 77)(73 88)(74 99)(75 110)(78 83)(79 94)(80 105)(81 116)(84 89)(85 100)(86 111)(90 95)(91 106)(92 117)(96 101)(97 112)(102 107)(103 118)(108 113)(114 119)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(61 67)(62 96)(63 65)(64 94)(66 92)(68 90)(69 119)(70 88)(71 117)(72 86)(73 115)(74 84)(75 113)(76 82)(77 111)(78 80)(79 109)(81 107)(83 105)(85 103)(87 101)(89 99)(91 97)(93 95)(98 120)(100 118)(102 116)(104 114)(106 112)(108 110)
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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79), (2,12)(3,23)(4,34)(5,45)(6,56)(8,18)(9,29)(10,40)(11,51)(14,24)(15,35)(16,46)(17,57)(20,30)(21,41)(22,52)(26,36)(27,47)(28,58)(32,42)(33,53)(38,48)(39,59)(44,54)(50,60)(61,76)(62,87)(63,98)(64,109)(65,120)(66,71)(67,82)(68,93)(69,104)(70,115)(72,77)(73,88)(74,99)(75,110)(78,83)(79,94)(80,105)(81,116)(84,89)(85,100)(86,111)(90,95)(91,106)(92,117)(96,101)(97,112)(102,107)(103,118)(108,113)(114,119), (2,30)(3,59)(4,28)(5,57)(6,26)(7,55)(8,24)(9,53)(10,22)(11,51)(12,20)(13,49)(14,18)(15,47)(17,45)(19,43)(21,41)(23,39)(25,37)(27,35)(29,33)(32,60)(34,58)(36,56)(38,54)(40,52)(42,50)(44,48)(61,67)(62,96)(63,65)(64,94)(66,92)(68,90)(69,119)(70,88)(71,117)(72,86)(73,115)(74,84)(75,113)(76,82)(77,111)(78,80)(79,109)(81,107)(83,105)(85,103)(87,101)(89,99)(91,97)(93,95)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110)>;
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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79), (2,12)(3,23)(4,34)(5,45)(6,56)(8,18)(9,29)(10,40)(11,51)(14,24)(15,35)(16,46)(17,57)(20,30)(21,41)(22,52)(26,36)(27,47)(28,58)(32,42)(33,53)(38,48)(39,59)(44,54)(50,60)(61,76)(62,87)(63,98)(64,109)(65,120)(66,71)(67,82)(68,93)(69,104)(70,115)(72,77)(73,88)(74,99)(75,110)(78,83)(79,94)(80,105)(81,116)(84,89)(85,100)(86,111)(90,95)(91,106)(92,117)(96,101)(97,112)(102,107)(103,118)(108,113)(114,119), (2,30)(3,59)(4,28)(5,57)(6,26)(7,55)(8,24)(9,53)(10,22)(11,51)(12,20)(13,49)(14,18)(15,47)(17,45)(19,43)(21,41)(23,39)(25,37)(27,35)(29,33)(32,60)(34,58)(36,56)(38,54)(40,52)(42,50)(44,48)(61,67)(62,96)(63,65)(64,94)(66,92)(68,90)(69,119)(70,88)(71,117)(72,86)(73,115)(74,84)(75,113)(76,82)(77,111)(78,80)(79,109)(81,107)(83,105)(85,103)(87,101)(89,99)(91,97)(93,95)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110) );
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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,120),(20,119),(21,118),(22,117),(23,116),(24,115),(25,114),(26,113),(27,112),(28,111),(29,110),(30,109),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,90),(50,89),(51,88),(52,87),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,80),(60,79)], [(2,12),(3,23),(4,34),(5,45),(6,56),(8,18),(9,29),(10,40),(11,51),(14,24),(15,35),(16,46),(17,57),(20,30),(21,41),(22,52),(26,36),(27,47),(28,58),(32,42),(33,53),(38,48),(39,59),(44,54),(50,60),(61,76),(62,87),(63,98),(64,109),(65,120),(66,71),(67,82),(68,93),(69,104),(70,115),(72,77),(73,88),(74,99),(75,110),(78,83),(79,94),(80,105),(81,116),(84,89),(85,100),(86,111),(90,95),(91,106),(92,117),(96,101),(97,112),(102,107),(103,118),(108,113),(114,119)], [(2,30),(3,59),(4,28),(5,57),(6,26),(7,55),(8,24),(9,53),(10,22),(11,51),(12,20),(13,49),(14,18),(15,47),(17,45),(19,43),(21,41),(23,39),(25,37),(27,35),(29,33),(32,60),(34,58),(36,56),(38,54),(40,52),(42,50),(44,48),(61,67),(62,96),(63,65),(64,94),(66,92),(68,90),(69,119),(70,88),(71,117),(72,86),(73,115),(74,84),(75,113),(76,82),(77,111),(78,80),(79,109),(81,107),(83,105),(85,103),(87,101),(89,99),(91,97),(93,95),(98,120),(100,118),(102,116),(104,114),(106,112),(108,110)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 12 | 20 | 30 | 60 | 2 | 2 | 4 | 30 | 2 | 2 | 2 | 40 | 12 | 20 | 2 | 2 | 24 | 24 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | Q8⋊3D6 | C2×S3×D5 | D40⋊C2 | D10⋊D6 | D60⋊C22 |
| kernel | D60⋊C22 | D30.5C4 | C3⋊D40 | C5⋊D24 | C3×Q8⋊D5 | C5×Q8⋊2S3 | C20⋊D6 | Q8⋊3D15 | Q8⋊D5 | Dic15 | D30 | Q8⋊2S3 | C5⋊2C8 | D20 | C5×Q8 | C3⋊C8 | D12 | C3×Q8 | C15 | C10 | Q8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D60⋊C22 ►in GL8(𝔽241)
| 240 | 189 | 240 | 189 | 0 | 0 | 0 | 0 |
| 52 | 52 | 52 | 52 | 0 | 0 | 0 | 0 |
| 1 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 189 | 189 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 190 | 2 | 139 |
| 0 | 0 | 0 | 0 | 51 | 51 | 102 | 102 |
| 0 | 0 | 0 | 0 | 240 | 51 | 240 | 51 |
| 0 | 0 | 0 | 0 | 190 | 190 | 190 | 190 |
| 38 | 28 | 203 | 213 | 0 | 0 | 0 | 0 |
| 221 | 203 | 20 | 38 | 0 | 0 | 0 | 0 |
| 165 | 185 | 203 | 213 | 0 | 0 | 0 | 0 |
| 40 | 76 | 20 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 237 | 29 |
| 0 | 0 | 0 | 0 | 0 | 0 | 66 | 4 |
| 0 | 0 | 0 | 0 | 239 | 135 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 52 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 51 | 240 |
G:=sub<GL(8,GF(241))| [240,52,1,189,0,0,0,0,189,52,52,189,0,0,0,0,240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,0,0,1,51,240,190,0,0,0,0,190,51,51,190,0,0,0,0,2,102,240,190,0,0,0,0,139,102,51,190],[38,221,165,40,0,0,0,0,28,203,185,76,0,0,0,0,203,20,203,20,0,0,0,0,213,38,213,38,0,0,0,0,0,0,0,0,0,0,239,33,0,0,0,0,0,0,135,2,0,0,0,0,237,66,0,0,0,0,0,0,29,4,0,0],[1,0,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,189,240,52,0,0,0,0,0,240,0,1,0,0,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240] >;
D60⋊C22 in GAP, Magma, Sage, TeX
D_{60}\rtimes C_2^2 % in TeX
G:=Group("D60:C2^2"); // GroupNames label
G:=SmallGroup(480,582);
// by ID
G=gap.SmallGroup(480,582);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,100,675,346,185,80,1356,18822]);
// 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^11,d*a*d=a^29,c*b*c=a^55*b,d*b*d=a^58*b,c*d=d*c>;
// generators/relations