metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊10C4, C60.211D4, C22.3D60, Dic30⋊10C4, M4(2)⋊4D15, C15⋊18C4≀C2, (C2×C6).4D20, C4.3(C4×D15), (C2×C30).1D4, C20.50(C4×S3), C60.80(C2×C4), C12.18(C4×D5), (C2×C10).4D12, (C2×C20).71D6, (C2×C4).37D30, C5⋊4(D12⋊C4), C3⋊2(D20⋊7C4), (C4×Dic15)⋊1C2, (C2×C12).72D10, C10.36(D6⋊C4), C4.29(C15⋊7D4), (C5×M4(2))⋊10S3, (C3×M4(2))⋊10D5, (C2×C60).57C22, D60⋊11C2.8C2, C12.108(C5⋊D4), C20.108(C3⋊D4), C30.78(C22⋊C4), (C15×M4(2))⋊20C2, C6.21(D10⋊C4), C2.11(D30⋊3C4), SmallGroup(480,185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊10C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a43b >
Subgroups: 596 in 88 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4×Dic3, C3×M4(2), C4○D12, Dic15, C60, D30, C2×C30, C4×Dic5, C5×M4(2), C4○D20, D12⋊C4, C120, Dic30, C4×D15, D60, C2×Dic15, C15⋊7D4, C2×C60, D20⋊7C4, C4×Dic15, C15×M4(2), D60⋊11C2, D60⋊10C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, D12⋊C4, C4×D15, D60, C15⋊7D4, D20⋊7C4, D30⋊3C4, D60⋊10C4
►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 89)(2 88)(3 87)(4 86)(5 85)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)
(1 16 31 46)(2 45 32 15)(3 14 33 44)(4 43 34 13)(5 12 35 42)(6 41 36 11)(7 10 37 40)(8 39 38 9)(17 60 47 30)(18 29 48 59)(19 58 49 28)(20 27 50 57)(21 56 51 26)(22 25 52 55)(23 54 53 24)(61 89)(62 118)(63 87)(64 116)(65 85)(66 114)(67 83)(68 112)(69 81)(70 110)(71 79)(72 108)(73 77)(74 106)(76 104)(78 102)(80 100)(82 98)(84 96)(86 94)(88 92)(91 119)(93 117)(95 115)(97 113)(99 111)(101 109)(103 107)
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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90), (1,16,31,46)(2,45,32,15)(3,14,33,44)(4,43,34,13)(5,12,35,42)(6,41,36,11)(7,10,37,40)(8,39,38,9)(17,60,47,30)(18,29,48,59)(19,58,49,28)(20,27,50,57)(21,56,51,26)(22,25,52,55)(23,54,53,24)(61,89)(62,118)(63,87)(64,116)(65,85)(66,114)(67,83)(68,112)(69,81)(70,110)(71,79)(72,108)(73,77)(74,106)(76,104)(78,102)(80,100)(82,98)(84,96)(86,94)(88,92)(91,119)(93,117)(95,115)(97,113)(99,111)(101,109)(103,107)>;
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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90), (1,16,31,46)(2,45,32,15)(3,14,33,44)(4,43,34,13)(5,12,35,42)(6,41,36,11)(7,10,37,40)(8,39,38,9)(17,60,47,30)(18,29,48,59)(19,58,49,28)(20,27,50,57)(21,56,51,26)(22,25,52,55)(23,54,53,24)(61,89)(62,118)(63,87)(64,116)(65,85)(66,114)(67,83)(68,112)(69,81)(70,110)(71,79)(72,108)(73,77)(74,106)(76,104)(78,102)(80,100)(82,98)(84,96)(86,94)(88,92)(91,119)(93,117)(95,115)(97,113)(99,111)(101,109)(103,107) );
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)], [(1,89),(2,88),(3,87),(4,86),(5,85),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,73),(18,72),(19,71),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,120),(31,119),(32,118),(33,117),(34,116),(35,115),(36,114),(37,113),(38,112),(39,111),(40,110),(41,109),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98),(53,97),(54,96),(55,95),(56,94),(57,93),(58,92),(59,91),(60,90)], [(1,16,31,46),(2,45,32,15),(3,14,33,44),(4,43,34,13),(5,12,35,42),(6,41,36,11),(7,10,37,40),(8,39,38,9),(17,60,47,30),(18,29,48,59),(19,58,49,28),(20,27,50,57),(21,56,51,26),(22,25,52,55),(23,54,53,24),(61,89),(62,118),(63,87),(64,116),(65,85),(66,114),(67,83),(68,112),(69,81),(70,110),(71,79),(72,108),(73,77),(74,106),(76,104),(78,102),(80,100),(82,98),(84,96),(86,94),(88,92),(91,119),(93,117),(95,115),(97,113),(99,111),(101,109),(103,107)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 60 | 2 | 1 | 1 | 2 | 30 | 30 | 30 | 30 | 60 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | D6 | D10 | C4×S3 | C3⋊D4 | D12 | D15 | C4≀C2 | C4×D5 | C5⋊D4 | D20 | D30 | C4×D15 | C15⋊7D4 | D60 | D12⋊C4 | D20⋊7C4 | D60⋊10C4 |
| kernel | D60⋊10C4 | C4×Dic15 | C15×M4(2) | D60⋊11C2 | Dic30 | D60 | C5×M4(2) | C60 | C2×C30 | C3×M4(2) | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | M4(2) | C15 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 4 | 8 |
Matrix representation of D60⋊10C4 ►in GL4(𝔽241) generated by
| 68 | 80 | 0 | 0 |
| 98 | 211 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 168 | 177 |
| 177 | 148 | 0 | 0 |
| 114 | 64 | 0 | 0 |
| 0 | 0 | 222 | 3 |
| 0 | 0 | 121 | 19 |
| 52 | 240 | 0 | 0 |
| 52 | 189 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 231 | 1 |
G:=sub<GL(4,GF(241))| [68,98,0,0,80,211,0,0,0,0,64,168,0,0,0,177],[177,114,0,0,148,64,0,0,0,0,222,121,0,0,3,19],[52,52,0,0,240,189,0,0,0,0,177,231,0,0,0,1] >;
D60⋊10C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{10}C_4 % in TeX
G:=Group("D60:10C4"); // GroupNames label
G:=SmallGroup(480,185);
// by ID
G=gap.SmallGroup(480,185);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,675,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^43*b>;
// generators/relations