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 [×4], C3, C4, C4 [×2], C22 [×5], C5, S3 [×2], C6, C6 [×2], C8, C2×C4 [×2], D4 [×3], C23, D5 [×2], D5, C10, C10, C12, C12 [×2], D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10 [×3], C2×C10, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C22×S3, C5×S3, C3×D5 [×2], 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 [×2], 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 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], 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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 120)(10 119)(11 118)(12 117)(13 116)(14 115)(15 114)(16 113)(17 112)(18 111)(19 110)(20 109)(21 108)(22 107)(23 106)(24 105)(25 104)(26 103)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)
(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)(61 118 97 70)(62 65 86 113)(63 72 75 96)(64 79)(66 93 102 105)(67 100 91 88)(68 107 80 71)(69 114)(73 82 85 106)(74 89)(76 103 112 115)(77 110 101 98)(78 117 90 81)(83 92 95 116)(84 99)(87 120 111 108)(94 109)(104 119)
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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,120)(10,119)(11,118)(12,117)(13,116)(14,115)(15,114)(16,113)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69), (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)(61,118,97,70)(62,65,86,113)(63,72,75,96)(64,79)(66,93,102,105)(67,100,91,88)(68,107,80,71)(69,114)(73,82,85,106)(74,89)(76,103,112,115)(77,110,101,98)(78,117,90,81)(83,92,95,116)(84,99)(87,120,111,108)(94,109)(104,119)>;
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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,120)(10,119)(11,118)(12,117)(13,116)(14,115)(15,114)(16,113)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69), (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)(61,118,97,70)(62,65,86,113)(63,72,75,96)(64,79)(66,93,102,105)(67,100,91,88)(68,107,80,71)(69,114)(73,82,85,106)(74,89)(76,103,112,115)(77,110,101,98)(78,117,90,81)(83,92,95,116)(84,99)(87,120,111,108)(94,109)(104,119) );
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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,120),(10,119),(11,118),(12,117),(13,116),(14,115),(15,114),(16,113),(17,112),(18,111),(19,110),(20,109),(21,108),(22,107),(23,106),(24,105),(25,104),(26,103),(27,102),(28,101),(29,100),(30,99),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,91),(39,90),(40,89),(41,88),(42,87),(43,86),(44,85),(45,84),(46,83),(47,82),(48,81),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69)], [(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),(61,118,97,70),(62,65,86,113),(63,72,75,96),(64,79),(66,93,102,105),(67,100,91,88),(68,107,80,71),(69,114),(73,82,85,106),(74,89),(76,103,112,115),(77,110,101,98),(78,117,90,81),(83,92,95,116),(84,99),(87,120,111,108),(94,109),(104,119)])
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