metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.81D4, C20.14D12, D12.31D10, D60⋊36C22, C60.128C23, Dic30⋊32C22, C5⋊2C8⋊3D6, (C2×D12)⋊9D5, C5⋊4(C8⋊D6), (C10×D12)⋊2C2, C5⋊D24⋊13C2, (C2×C20).90D6, (C2×C30).47D4, C30.79(C2×D4), C4.Dic5⋊7S3, C15⋊10(C8⋊C22), C10.49(C2×D12), (C2×C10).39D12, (C2×C12).90D10, D60⋊11C2⋊2C2, D12.D5⋊13C2, C3⋊1(D4.D10), C12.27(C5⋊D4), C4.23(C5⋊D12), C20.90(C22×S3), (C2×C60).25C22, (C5×D12).36C22, C12.151(C22×D5), C22.4(C5⋊D12), C4.76(C2×S3×D5), C6.3(C2×C5⋊D4), (C2×C4).10(S3×D5), C2.7(C2×C5⋊D12), (C3×C4.Dic5)⋊2C2, (C3×C5⋊2C8)⋊17C22, (C2×C6).11(C5⋊D4), SmallGroup(480,380)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊36C22
G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a11, ad=da, cbc=a25b, dbd=a30b, cd=dc >
Subgroups: 764 in 136 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C22×S3, C5×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C22×C10, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, Dic15, C60, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, D4.D5, C4○D20, D4×C10, C8⋊D6, C3×C5⋊2C8, C5×D12, C5×D12, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, S3×C2×C10, D4.D10, C5⋊D24, D12.D5, C3×C4.Dic5, C10×D12, D60⋊11C2, D60⋊36C22
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C8⋊C22, C5⋊D4, C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8⋊D6, C5⋊D12, C2×S3×D5, D4.D10, C2×C5⋊D12, D60⋊36C22
►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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)
(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)(61 116)(62 67)(63 78)(64 89)(65 100)(66 111)(68 73)(69 84)(70 95)(71 106)(72 117)(74 79)(75 90)(76 101)(77 112)(80 85)(81 96)(82 107)(83 118)(86 91)(87 102)(88 113)(92 97)(93 108)(94 119)(98 103)(99 114)(104 109)(105 120)(110 115)
(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78), (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)(61,116)(62,67)(63,78)(64,89)(65,100)(66,111)(68,73)(69,84)(70,95)(71,106)(72,117)(74,79)(75,90)(76,101)(77,112)(80,85)(81,96)(82,107)(83,118)(86,91)(87,102)(88,113)(92,97)(93,108)(94,119)(98,103)(99,114)(104,109)(105,120)(110,115), (61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78), (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)(61,116)(62,67)(63,78)(64,89)(65,100)(66,111)(68,73)(69,84)(70,95)(71,106)(72,117)(74,79)(75,90)(76,101)(77,112)(80,85)(81,96)(82,107)(83,118)(86,91)(87,102)(88,113)(92,97)(93,108)(94,119)(98,103)(99,114)(104,109)(105,120)(110,115), (61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,120),(19,119),(20,118),(21,117),(22,116),(23,115),(24,114),(25,113),(26,112),(27,111),(28,110),(29,109),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78)], [(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),(61,116),(62,67),(63,78),(64,89),(65,100),(66,111),(68,73),(69,84),(70,95),(71,106),(72,117),(74,79),(75,90),(76,101),(77,112),(80,85),(81,96),(82,107),(83,118),(86,91),(87,102),(88,113),(92,97),(93,108),(94,119),(98,103),(99,114),(104,109),(105,120),(110,115)], [(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 12 | 60 | 2 | 2 | 2 | 60 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D12 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D5 | C8⋊D6 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | D4.D10 | D60⋊36C22 |
| kernel | D60⋊36C22 | C5⋊D24 | D12.D5 | C3×C4.Dic5 | C10×D12 | D60⋊11C2 | C4.Dic5 | C60 | C2×C30 | C2×D12 | C5⋊2C8 | C2×C20 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D60⋊36C22 ►in GL4(𝔽241) generated by
| 55 | 27 | 0 | 0 |
| 162 | 0 | 0 | 0 |
| 0 | 0 | 206 | 180 |
| 0 | 0 | 116 | 0 |
| 0 | 0 | 0 | 180 |
| 0 | 0 | 125 | 0 |
| 0 | 27 | 0 | 0 |
| 79 | 0 | 0 | 0 |
| 1 | 19 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 56 | 194 |
| 0 | 0 | 159 | 185 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [55,162,0,0,27,0,0,0,0,0,206,116,0,0,180,0],[0,0,0,79,0,0,27,0,0,125,0,0,180,0,0,0],[1,0,0,0,19,240,0,0,0,0,56,159,0,0,194,185],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;
D60⋊36C22 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{36}C_2^2 % in TeX
G:=Group("D60:36C2^2"); // GroupNames label
G:=SmallGroup(480,380);
// by ID
G=gap.SmallGroup(480,380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,346,80,1356,18822]);
// 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,a*d=d*a,c*b*c=a^25*b,d*b*d=a^30*b,c*d=d*c>;
// generators/relations