metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊1C4, D12⋊3F5, Dic5.1D12, C4⋊F5⋊1S3, C5⋊(C6.D8), C4.8(S3×F5), (C5×D12)⋊1C4, C20.1(C4×S3), C60.1(C2×C4), (C3×D5).1D8, C12.1(C2×F5), C3⋊2(D20⋊C4), (C4×D5).19D6, (C6×D5).17D4, (D5×D12).3C2, C2.4(D6⋊F5), C60.C4⋊1C2, C15⋊1(D4⋊C4), D5.2(D4⋊S3), C10.1(D6⋊C4), (C3×D5).1SD16, C6.1(C22⋊F5), C30.1(C22⋊C4), (C3×Dic5).20D4, D10.22(C3⋊D4), D5.2(Q8⋊2S3), (D5×C12).30C22, (C3×C4⋊F5)⋊1C2, SmallGroup(480,227)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a43, cbc-1=a57b >
Subgroups: 788 in 100 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, D5, C10, C10, C12, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C3⋊C8, D12, D12, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60, C3×F5, S3×D5, C6×D5, S3×C10, D30, D5⋊C8, C4⋊F5, D4×D5, C6.D8, C15⋊C8, C5⋊D12, D5×C12, C5×D12, D60, C6×F5, C2×S3×D5, D20⋊C4, C3×C4⋊F5, C60.C4, D5×D12, D60⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, F5, C4×S3, D12, C3⋊D4, D4⋊C4, C2×F5, D6⋊C4, D4⋊S3, Q8⋊2S3, C22⋊F5, C6.D8, S3×F5, D20⋊C4, D6⋊F5, D60⋊C4
►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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 112)(22 111)(23 110)(24 109)(25 108)(26 107)(27 106)(28 105)(29 104)(30 103)(31 102)(32 101)(33 100)(34 99)(35 98)(36 97)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)
(2 8 50 44)(3 15 39 27)(4 22 28 10)(5 29 17 53)(6 36)(7 43 55 19)(9 57 33 45)(12 18 60 54)(13 25 49 37)(14 32 38 20)(16 46)(23 35 59 47)(24 42 48 30)(26 56)(34 52 58 40)(61 94 85 82)(62 101 74 65)(63 108)(64 115 112 91)(66 69 90 117)(67 76 79 100)(68 83)(70 97 106 109)(71 104 95 92)(72 111 84 75)(73 118)(77 86 89 110)(78 93)(80 107 116 119)(81 114 105 102)(87 96 99 120)(88 103)(98 113)
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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73), (2,8,50,44)(3,15,39,27)(4,22,28,10)(5,29,17,53)(6,36)(7,43,55,19)(9,57,33,45)(12,18,60,54)(13,25,49,37)(14,32,38,20)(16,46)(23,35,59,47)(24,42,48,30)(26,56)(34,52,58,40)(61,94,85,82)(62,101,74,65)(63,108)(64,115,112,91)(66,69,90,117)(67,76,79,100)(68,83)(70,97,106,109)(71,104,95,92)(72,111,84,75)(73,118)(77,86,89,110)(78,93)(80,107,116,119)(81,114,105,102)(87,96,99,120)(88,103)(98,113)>;
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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73), (2,8,50,44)(3,15,39,27)(4,22,28,10)(5,29,17,53)(6,36)(7,43,55,19)(9,57,33,45)(12,18,60,54)(13,25,49,37)(14,32,38,20)(16,46)(23,35,59,47)(24,42,48,30)(26,56)(34,52,58,40)(61,94,85,82)(62,101,74,65)(63,108)(64,115,112,91)(66,69,90,117)(67,76,79,100)(68,83)(70,97,106,109)(71,104,95,92)(72,111,84,75)(73,118)(77,86,89,110)(78,93)(80,107,116,119)(81,114,105,102)(87,96,99,120)(88,103)(98,113) );
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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,120),(14,119),(15,118),(16,117),(17,116),(18,115),(19,114),(20,113),(21,112),(22,111),(23,110),(24,109),(25,108),(26,107),(27,106),(28,105),(29,104),(30,103),(31,102),(32,101),(33,100),(34,99),(35,98),(36,97),(37,96),(38,95),(39,94),(40,93),(41,92),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85),(49,84),(50,83),(51,82),(52,81),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73)], [(2,8,50,44),(3,15,39,27),(4,22,28,10),(5,29,17,53),(6,36),(7,43,55,19),(9,57,33,45),(12,18,60,54),(13,25,49,37),(14,32,38,20),(16,46),(23,35,59,47),(24,42,48,30),(26,56),(34,52,58,40),(61,94,85,82),(62,101,74,65),(63,108),(64,115,112,91),(66,69,90,117),(67,76,79,100),(68,83),(70,97,106,109),(71,104,95,92),(72,111,84,75),(73,118),(77,86,89,110),(78,93),(80,107,116,119),(81,114,105,102),(87,96,99,120),(88,103),(98,113)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 12A | 12B | ··· | 12F | 15 | 20 | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | ··· | 12 | 15 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 12 | 60 | 2 | 2 | 10 | 20 | 20 | 4 | 2 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 24 | 24 | 4 | 20 | ··· | 20 | 8 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | D12 | C4×S3 | C3⋊D4 | F5 | C2×F5 | D4⋊S3 | Q8⋊2S3 | C22⋊F5 | S3×F5 | D20⋊C4 | D6⋊F5 | D60⋊C4 |
| kernel | D60⋊C4 | C3×C4⋊F5 | C60.C4 | D5×D12 | C5×D12 | D60 | C4⋊F5 | C3×Dic5 | C6×D5 | C4×D5 | C3×D5 | C3×D5 | Dic5 | C20 | D10 | D12 | C12 | D5 | D5 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D60⋊C4 ►in GL8(𝔽241)
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 116 | 232 | 125 | 9 | 0 | 0 | 0 | 0 |
| 116 | 125 | 125 | 116 | 0 | 0 | 0 | 0 |
| 125 | 9 | 125 | 9 | 0 | 0 | 0 | 0 |
| 125 | 116 | 125 | 116 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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))| [0,0,1,240,0,0,0,0,0,0,1,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240,0],[116,116,125,125,0,0,0,0,232,125,9,116,0,0,0,0,125,125,125,125,0,0,0,0,9,116,9,116,0,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,240,0,1,0,0,0,0,240,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240] >;
D60⋊C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes C_4 % in TeX
G:=Group("D60:C4"); // GroupNames label
G:=SmallGroup(480,227);
// by ID
G=gap.SmallGroup(480,227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^57*b>;
// generators/relations