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 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, C4×S3 [×2], D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C8⋊C22, C5⋊2C8, C40, C4×D5 [×2], D20, D20 [×2], 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 [×2], 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 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], 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)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)
(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 96)(62 107)(63 118)(64 69)(65 80)(66 91)(67 102)(68 113)(70 75)(71 86)(72 97)(73 108)(74 119)(76 81)(77 92)(78 103)(79 114)(82 87)(83 98)(84 109)(85 120)(88 93)(89 104)(90 115)(94 99)(95 110)(100 105)(101 116)(106 111)(112 117)
(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 75)(62 104)(63 73)(64 102)(65 71)(66 100)(67 69)(68 98)(70 96)(72 94)(74 92)(76 90)(77 119)(78 88)(79 117)(80 86)(81 115)(82 84)(83 113)(85 111)(87 109)(89 107)(91 105)(93 103)(95 101)(97 99)(106 120)(108 118)(110 116)(112 114)
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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113), (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)(61,96)(62,107)(63,118)(64,69)(65,80)(66,91)(67,102)(68,113)(70,75)(71,86)(72,97)(73,108)(74,119)(76,81)(77,92)(78,103)(79,114)(82,87)(83,98)(84,109)(85,120)(88,93)(89,104)(90,115)(94,99)(95,110)(100,105)(101,116)(106,111)(112,117), (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)(61,75)(62,104)(63,73)(64,102)(65,71)(66,100)(67,69)(68,98)(70,96)(72,94)(74,92)(76,90)(77,119)(78,88)(79,117)(80,86)(81,115)(82,84)(83,113)(85,111)(87,109)(89,107)(91,105)(93,103)(95,101)(97,99)(106,120)(108,118)(110,116)(112,114)>;
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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113), (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)(61,96)(62,107)(63,118)(64,69)(65,80)(66,91)(67,102)(68,113)(70,75)(71,86)(72,97)(73,108)(74,119)(76,81)(77,92)(78,103)(79,114)(82,87)(83,98)(84,109)(85,120)(88,93)(89,104)(90,115)(94,99)(95,110)(100,105)(101,116)(106,111)(112,117), (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)(61,75)(62,104)(63,73)(64,102)(65,71)(66,100)(67,69)(68,98)(70,96)(72,94)(74,92)(76,90)(77,119)(78,88)(79,117)(80,86)(81,115)(82,84)(83,113)(85,111)(87,109)(89,107)(91,105)(93,103)(95,101)(97,99)(106,120)(108,118)(110,116)(112,114) );
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)], [(1,112),(2,111),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,102),(12,101),(13,100),(14,99),(15,98),(16,97),(17,96),(18,95),(19,94),(20,93),(21,92),(22,91),(23,90),(24,89),(25,88),(26,87),(27,86),(28,85),(29,84),(30,83),(31,82),(32,81),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,120),(54,119),(55,118),(56,117),(57,116),(58,115),(59,114),(60,113)], [(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),(61,96),(62,107),(63,118),(64,69),(65,80),(66,91),(67,102),(68,113),(70,75),(71,86),(72,97),(73,108),(74,119),(76,81),(77,92),(78,103),(79,114),(82,87),(83,98),(84,109),(85,120),(88,93),(89,104),(90,115),(94,99),(95,110),(100,105),(101,116),(106,111),(112,117)], [(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),(61,75),(62,104),(63,73),(64,102),(65,71),(66,100),(67,69),(68,98),(70,96),(72,94),(74,92),(76,90),(77,119),(78,88),(79,117),(80,86),(81,115),(82,84),(83,113),(85,111),(87,109),(89,107),(91,105),(93,103),(95,101),(97,99),(106,120),(108,118),(110,116),(112,114)])
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