metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊5C4, Dic6⋊2F5, Dic5.21D12, C15⋊4C4≀C2, (C4×F5)⋊2S3, C4.4(S3×F5), C20.8(C4×S3), (C12×F5)⋊2C2, C60.11(C2×C4), (C5×Dic6)⋊5C4, (C6×D5).24D4, C3⋊2(Q8⋊2F5), (C4×D5).26D6, C12.25(C2×F5), C12.F5⋊2C2, C5⋊1(C42⋊4S3), C10.8(D6⋊C4), C2.11(D6⋊F5), C6.8(C22⋊F5), C12.28D10.5C2, D10.2(C3⋊D4), C30.8(C22⋊C4), (C3×Dic5).27D4, (D5×C12).41C22, SmallGroup(480,234)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊5C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a57b >
Subgroups: 564 in 88 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, F5, D10, D10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D5, D15, C30, C4≀C2, C5⋊C8, C4×D5, C4×D5, D20, C5×Q8, C2×F5, C4.Dic3, C4×C12, C4○D12, C5×Dic3, C3×Dic5, C60, C3×F5, C6×D5, D30, C4.F5, C4×F5, Q8⋊2D5, C42⋊4S3, C15⋊C8, D30.C2, C3⋊D20, D5×C12, C5×Dic6, D60, C6×F5, Q8⋊2F5, C12×F5, C12.F5, C12.28D10, D60⋊5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C4≀C2, C2×F5, D6⋊C4, C22⋊F5, C42⋊4S3, S3×F5, Q8⋊2F5, D6⋊F5, D60⋊5C4
►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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 120)(21 119)(22 118)(23 117)(24 116)(25 115)(26 114)(27 113)(28 112)(29 111)(30 110)(31 109)(32 108)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 100)(41 99)(42 98)(43 97)(44 96)(45 95)(46 94)(47 93)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 80)
(2 38 50 14)(3 15 39 27)(4 52 28 40)(5 29 17 53)(7 43 55 19)(8 20 44 32)(9 57 33 45)(10 34 22 58)(12 48 60 24)(13 25 49 37)(18 30 54 42)(23 35 59 47)(61 82 79 88)(62 119 68 101)(63 96 117 114)(64 73 106 67)(65 110 95 80)(66 87 84 93)(69 78 111 72)(70 115 100 85)(71 92 89 98)(74 83 116 77)(75 120 105 90)(76 97 94 103)(81 102 99 108)(86 107 104 113)(91 112 109 118)
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,120)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,112)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,100)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80), (2,38,50,14)(3,15,39,27)(4,52,28,40)(5,29,17,53)(7,43,55,19)(8,20,44,32)(9,57,33,45)(10,34,22,58)(12,48,60,24)(13,25,49,37)(18,30,54,42)(23,35,59,47)(61,82,79,88)(62,119,68,101)(63,96,117,114)(64,73,106,67)(65,110,95,80)(66,87,84,93)(69,78,111,72)(70,115,100,85)(71,92,89,98)(74,83,116,77)(75,120,105,90)(76,97,94,103)(81,102,99,108)(86,107,104,113)(91,112,109,118)>;
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,120)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,112)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,100)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80), (2,38,50,14)(3,15,39,27)(4,52,28,40)(5,29,17,53)(7,43,55,19)(8,20,44,32)(9,57,33,45)(10,34,22,58)(12,48,60,24)(13,25,49,37)(18,30,54,42)(23,35,59,47)(61,82,79,88)(62,119,68,101)(63,96,117,114)(64,73,106,67)(65,110,95,80)(66,87,84,93)(69,78,111,72)(70,115,100,85)(71,92,89,98)(74,83,116,77)(75,120,105,90)(76,97,94,103)(81,102,99,108)(86,107,104,113)(91,112,109,118) );
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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,120),(21,119),(22,118),(23,117),(24,116),(25,115),(26,114),(27,113),(28,112),(29,111),(30,110),(31,109),(32,108),(33,107),(34,106),(35,105),(36,104),(37,103),(38,102),(39,101),(40,100),(41,99),(42,98),(43,97),(44,96),(45,95),(46,94),(47,93),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,84),(57,83),(58,82),(59,81),(60,80)], [(2,38,50,14),(3,15,39,27),(4,52,28,40),(5,29,17,53),(7,43,55,19),(8,20,44,32),(9,57,33,45),(10,34,22,58),(12,48,60,24),(13,25,49,37),(18,30,54,42),(23,35,59,47),(61,82,79,88),(62,119,68,101),(63,96,117,114),(64,73,106,67),(65,110,95,80),(66,87,84,93),(69,78,111,72),(70,115,100,85),(71,92,89,98),(74,83,116,77),(75,120,105,90),(76,97,94,103),(81,102,99,108),(86,107,104,113),(91,112,109,118)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | 8B | 10 | 12A | 12B | 12C | ··· | 12L | 15 | 20A | 20B | 20C | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 12 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 10 | 60 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 10 | 12 | 4 | 2 | 10 | 10 | 60 | 60 | 4 | 2 | 2 | 10 | ··· | 10 | 8 | 8 | 24 | 24 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D12 | C4×S3 | C3⋊D4 | C4≀C2 | C42⋊4S3 | F5 | C2×F5 | C22⋊F5 | S3×F5 | Q8⋊2F5 | D6⋊F5 | D60⋊5C4 |
| kernel | D60⋊5C4 | C12×F5 | C12.F5 | C12.28D10 | C5×Dic6 | D60 | C4×F5 | C3×Dic5 | C6×D5 | C4×D5 | Dic5 | C20 | D10 | C15 | C5 | Dic6 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D60⋊5C4 ►in GL8(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 236 | 0 | 0 | 0 | 0 |
| 0 | 0 | 97 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 159 | 142 | 0 | 0 | 0 | 0 | 0 | 0 |
| 46 | 82 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 182 | 0 | 0 | 0 | 0 |
| 0 | 0 | 237 | 214 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 234 | 117 | 234 |
| 0 | 0 | 0 | 0 | 234 | 117 | 234 | 0 |
| 0 | 0 | 0 | 0 | 124 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 117 | 124 | 124 | 117 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 119 | 177 | 0 | 0 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
G:=sub<GL(8,GF(241))| [64,23,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,239,97,0,0,0,0,0,0,236,1,0,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0,0,0,0,0,240,0,1,0,0,0,0,0,0,240,1,0],[159,46,0,0,0,0,0,0,142,82,0,0,0,0,0,0,0,0,27,237,0,0,0,0,0,0,182,214,0,0,0,0,0,0,0,0,0,234,124,117,0,0,0,0,234,117,0,124,0,0,0,0,117,234,7,124,0,0,0,0,234,0,7,117],[1,119,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240] >;
D60⋊5C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_5C_4 % in TeX
G:=Group("D60:5C4"); // GroupNames label
G:=SmallGroup(480,234);
// by ID
G=gap.SmallGroup(480,234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^57*b>;
// generators/relations