metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊2C4, Dic6⋊4F5, D10.2D12, C15⋊3C4≀C2, C4.F5⋊2S3, C20.7(C4×S3), C60.4(C2×C4), C12.4(C2×F5), C4.11(S3×F5), (C5×Dic6)⋊2C4, (C4×D5).25D6, C3⋊1(Q8⋊2F5), (C6×D5).23D4, C5⋊1(D12⋊C4), C10.7(D6⋊C4), C2.10(D6⋊F5), C6.7(C22⋊F5), C12.28D10.3C2, C30.7(C22⋊C4), (C3×Dic5).26D4, (D5×C12).33C22, Dic5.27(C3⋊D4), (C4×C3⋊F5)⋊2C2, (C3×C4.F5)⋊2C2, SmallGroup(480,233)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊2C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a17, cbc-1=a31b >
Subgroups: 612 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, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D5, D15, C30, C4≀C2, C5⋊C8, C4×D5, C4×D5, D20, C5×Q8, C2×F5, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, C60, C3⋊F5, C6×D5, D30, C4.F5, C4×F5, Q8⋊2D5, D12⋊C4, C3×C5⋊C8, D30.C2, C3⋊D20, D5×C12, C5×Dic6, D60, C2×C3⋊F5, Q8⋊2F5, C3×C4.F5, C4×C3⋊F5, C12.28D10, D60⋊2C4
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, D12⋊C4, S3×F5, Q8⋊2F5, D6⋊F5, D60⋊2C4
►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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 84)(23 83)(24 82)(25 81)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)
(1 16 31 46)(2 9 20 3)(4 55 58 37)(5 48 47 54)(6 41 36 11)(7 34 25 28)(8 27 14 45)(10 13 52 19)(12 59 30 53)(15 38 57 44)(17 24 35 18)(21 56 51 26)(22 49 40 43)(23 42 29 60)(32 39 50 33)(62 114 110 78)(63 107 99 95)(64 100 88 112)(65 93 77 69)(66 86)(67 79 115 103)(68 72 104 120)(70 118 82 94)(71 111)(73 97 109 85)(74 90 98 102)(75 83 87 119)(80 108 92 84)(81 101)(89 105 113 117)(96 116)
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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106), (1,16,31,46)(2,9,20,3)(4,55,58,37)(5,48,47,54)(6,41,36,11)(7,34,25,28)(8,27,14,45)(10,13,52,19)(12,59,30,53)(15,38,57,44)(17,24,35,18)(21,56,51,26)(22,49,40,43)(23,42,29,60)(32,39,50,33)(62,114,110,78)(63,107,99,95)(64,100,88,112)(65,93,77,69)(66,86)(67,79,115,103)(68,72,104,120)(70,118,82,94)(71,111)(73,97,109,85)(74,90,98,102)(75,83,87,119)(80,108,92,84)(81,101)(89,105,113,117)(96,116)>;
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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106), (1,16,31,46)(2,9,20,3)(4,55,58,37)(5,48,47,54)(6,41,36,11)(7,34,25,28)(8,27,14,45)(10,13,52,19)(12,59,30,53)(15,38,57,44)(17,24,35,18)(21,56,51,26)(22,49,40,43)(23,42,29,60)(32,39,50,33)(62,114,110,78)(63,107,99,95)(64,100,88,112)(65,93,77,69)(66,86)(67,79,115,103)(68,72,104,120)(70,118,82,94)(71,111)(73,97,109,85)(74,90,98,102)(75,83,87,119)(80,108,92,84)(81,101)(89,105,113,117)(96,116) );
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)], [(1,105),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,84),(23,83),(24,82),(25,81),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,73),(34,72),(35,71),(36,70),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,120),(47,119),(48,118),(49,117),(50,116),(51,115),(52,114),(53,113),(54,112),(55,111),(56,110),(57,109),(58,108),(59,107),(60,106)], [(1,16,31,46),(2,9,20,3),(4,55,58,37),(5,48,47,54),(6,41,36,11),(7,34,25,28),(8,27,14,45),(10,13,52,19),(12,59,30,53),(15,38,57,44),(17,24,35,18),(21,56,51,26),(22,49,40,43),(23,42,29,60),(32,39,50,33),(62,114,110,78),(63,107,99,95),(64,100,88,112),(65,93,77,69),(66,86),(67,79,115,103),(68,72,104,120),(70,118,82,94),(71,111),(73,97,109,85),(74,90,98,102),(75,83,87,119),(80,108,92,84),(81,101),(89,105,113,117),(96,116)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 8A | 8B | 10 | 12A | 12B | 12C | 15 | 20A | 20B | 20C | 24A | 24B | 24C | 24D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 10 | 60 | 2 | 2 | 5 | 5 | 12 | 30 | 30 | 30 | 30 | 4 | 2 | 20 | 20 | 20 | 4 | 4 | 10 | 10 | 8 | 8 | 24 | 24 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | C3⋊D4 | C4×S3 | D12 | C4≀C2 | F5 | C2×F5 | C22⋊F5 | D12⋊C4 | S3×F5 | Q8⋊2F5 | D6⋊F5 | D60⋊2C4 |
| kernel | D60⋊2C4 | C3×C4.F5 | C4×C3⋊F5 | C12.28D10 | C5×Dic6 | D60 | C4.F5 | C3×Dic5 | C6×D5 | C4×D5 | Dic5 | C20 | D10 | C15 | Dic6 | C12 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D60⋊2C4 ►in GL8(𝔽241)
| 177 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 240 | 1 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(241))| [177,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;
D60⋊2C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_2C_4 % in TeX
G:=Group("D60:2C4"); // GroupNames label
G:=SmallGroup(480,233);
// by ID
G=gap.SmallGroup(480,233);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,100,675,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^31*b>;
// generators/relations