metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.84D4, D20.32D6, C12.15D20, D60⋊30C22, C60.105C23, Dic30⋊27C22, C3⋊C8⋊4D10, (C6×D20)⋊2C2, (C2×D20)⋊9S3, C3⋊D40⋊14C2, C3⋊4(C8⋊D10), C6.50(C2×D20), (C2×C30).55D4, (C2×C6).40D20, C30.87(C2×D4), (C2×C20).96D6, C4.Dic3⋊8D5, C15⋊12(C8⋊C22), (C2×C12).98D10, C5⋊1(D12⋊6C22), D60⋊11C2⋊5C2, C6.D20⋊13C2, C4.23(C3⋊D20), C20.29(C3⋊D4), (C2×C60).33C22, C12.96(C22×D5), C20.155(C22×S3), (C3×D20).37C22, C22.4(C3⋊D20), C4.104(C2×S3×D5), (C5×C3⋊C8)⋊18C22, (C2×C4).14(S3×D5), C2.9(C2×C3⋊D20), C10.5(C2×C3⋊D4), (C5×C4.Dic3)⋊2C2, (C2×C10).12(C3⋊D4), SmallGroup(480,388)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊30C22
G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a19, ad=da, cbc=a3b, dbd=a30b, cd=dc >
Subgroups: 860 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, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×C6, C3×D5, D15, C30, C30, C8⋊C22, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C22×D5, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, Dic15, C60, C6×D5, D30, C2×C30, C40⋊C2, D40, C5×M4(2), C2×D20, C4○D20, D12⋊6C22, C5×C3⋊C8, C3×D20, C3×D20, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D5×C2×C6, C8⋊D10, C3⋊D40, C6.D20, C5×C4.Dic3, C6×D20, D60⋊11C2, D60⋊30C22
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8⋊C22, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, D12⋊6C22, C3⋊D20, C2×S3×D5, C8⋊D10, C2×C3⋊D20, D60⋊30C22
►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)
(1 120)(2 119)(3 118)(4 117)(5 116)(6 115)(7 114)(8 113)(9 112)(10 111)(11 110)(12 109)(13 108)(14 107)(15 106)(16 105)(17 104)(18 103)(19 102)(20 101)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
(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 76)(62 95)(63 114)(64 73)(65 92)(66 111)(67 70)(68 89)(69 108)(71 86)(72 105)(74 83)(75 102)(77 80)(78 99)(79 118)(81 96)(82 115)(84 93)(85 112)(87 90)(88 109)(91 106)(94 103)(97 100)(98 119)(101 116)(104 113)(107 110)(117 120)
(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), (1,120)(2,119)(3,118)(4,117)(5,116)(6,115)(7,114)(8,113)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61), (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)(61,76)(62,95)(63,114)(64,73)(65,92)(66,111)(67,70)(68,89)(69,108)(71,86)(72,105)(74,83)(75,102)(77,80)(78,99)(79,118)(81,96)(82,115)(84,93)(85,112)(87,90)(88,109)(91,106)(94,103)(97,100)(98,119)(101,116)(104,113)(107,110)(117,120), (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), (1,120)(2,119)(3,118)(4,117)(5,116)(6,115)(7,114)(8,113)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61), (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)(61,76)(62,95)(63,114)(64,73)(65,92)(66,111)(67,70)(68,89)(69,108)(71,86)(72,105)(74,83)(75,102)(77,80)(78,99)(79,118)(81,96)(82,115)(84,93)(85,112)(87,90)(88,109)(91,106)(94,103)(97,100)(98,119)(101,116)(104,113)(107,110)(117,120), (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)], [(1,120),(2,119),(3,118),(4,117),(5,116),(6,115),(7,114),(8,113),(9,112),(10,111),(11,110),(12,109),(13,108),(14,107),(15,106),(16,105),(17,104),(18,103),(19,102),(20,101),(21,100),(22,99),(23,98),(24,97),(25,96),(26,95),(27,94),(28,93),(29,92),(30,91),(31,90),(32,89),(33,88),(34,87),(35,86),(36,85),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61)], [(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),(61,76),(62,95),(63,114),(64,73),(65,92),(66,111),(67,70),(68,89),(69,108),(71,86),(72,105),(74,83),(75,102),(77,80),(78,99),(79,118),(81,96),(82,115),(84,93),(85,112),(87,90),(88,109),(91,106),(94,103),(97,100),(98,119),(101,116),(104,113),(107,110),(117,120)], [(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 | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 20 | 60 | 2 | 2 | 2 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 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 | C3⋊D4 | C3⋊D4 | D20 | D20 | C8⋊C22 | S3×D5 | D12⋊6C22 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | C8⋊D10 | D60⋊30C22 |
| kernel | D60⋊30C22 | C3⋊D40 | C6.D20 | C5×C4.Dic3 | C6×D20 | D60⋊11C2 | C2×D20 | C60 | C2×C30 | C4.Dic3 | D20 | C2×C20 | C3⋊C8 | 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⋊30C22 ►in GL4(𝔽241) generated by
| 98 | 98 | 0 | 0 |
| 143 | 133 | 0 | 0 |
| 0 | 0 | 24 | 24 |
| 0 | 0 | 217 | 67 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 132 | 16 |
| 15 | 0 | 0 | 0 |
| 57 | 226 | 0 | 0 |
| 1 | 189 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 200 | 119 |
| 0 | 0 | 156 | 41 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [98,143,0,0,98,133,0,0,0,0,24,217,0,0,24,67],[0,0,15,57,0,0,0,226,225,132,0,0,0,16,0,0],[1,0,0,0,189,240,0,0,0,0,200,156,0,0,119,41],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;
D60⋊30C22 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{30}C_2^2 % in TeX
G:=Group("D60:30C2^2"); // GroupNames label
G:=SmallGroup(480,388);
// by ID
G=gap.SmallGroup(480,388);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,675,80,1356,18822]);
// 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,a*d=d*a,c*b*c=a^3*b,d*b*d=a^30*b,c*d=d*c>;
// generators/relations