metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊13C4, C12.57D20, C60.100D4, Dic30⋊13C4, C15⋊9C4≀C2, C12.9(C4×D5), C20.65(C4×S3), C60.95(C2×C4), (C4×Dic3)⋊2D5, (C2×C30).27D4, C4.Dic5⋊1S3, C3⋊1(D20⋊4C4), C5⋊3(D12⋊C4), (Dic3×C20)⋊2C2, (C2×C20).309D6, (C2×C12).58D10, (C2×C10).33D12, C10.23(D6⋊C4), C4.29(C3⋊D20), C20.62(C3⋊D4), C4.3(D30.C2), (C2×C60).37C22, D60⋊11C2.4C2, C6.8(D10⋊C4), C2.9(D30⋊4C4), C30.63(C22⋊C4), C22.2(C5⋊D12), (C2×C4).87(S3×D5), (C2×C6).3(C5⋊D4), (C3×C4.Dic5)⋊4C2, SmallGroup(480,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊13C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a41, cbc-1=a55b >
Subgroups: 508 in 88 conjugacy classes, 32 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, C20, D10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C4≀C2, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, Dic15, C60, D30, C2×C30, C4.Dic5, C4×C20, C4○D20, D12⋊C4, C3×C5⋊2C8, C10×Dic3, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D20⋊4C4, C3×C4.Dic5, Dic3×C20, D60⋊11C2, D60⋊13C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, D12⋊C4, D30.C2, C3⋊D20, C5⋊D12, D20⋊4C4, D30⋊4C4, D60⋊13C4
►On 120 points
(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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 101)(31 100)(32 99)(33 98)(34 97)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)
(1 46 31 16)(2 27 32 57)(3 8 33 38)(4 49 34 19)(5 30 35 60)(6 11 36 41)(7 52 37 22)(9 14 39 44)(10 55 40 25)(12 17 42 47)(13 58 43 28)(15 20 45 50)(18 23 48 53)(21 26 51 56)(24 29 54 59)(61 111)(62 92)(63 73)(64 114)(65 95)(66 76)(67 117)(68 98)(69 79)(70 120)(71 101)(72 82)(74 104)(75 85)(77 107)(78 88)(80 110)(81 91)(83 113)(84 94)(86 116)(87 97)(89 119)(90 100)(93 103)(96 106)(99 109)(102 112)(105 115)(108 118)
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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71), (1,46,31,16)(2,27,32,57)(3,8,33,38)(4,49,34,19)(5,30,35,60)(6,11,36,41)(7,52,37,22)(9,14,39,44)(10,55,40,25)(12,17,42,47)(13,58,43,28)(15,20,45,50)(18,23,48,53)(21,26,51,56)(24,29,54,59)(61,111)(62,92)(63,73)(64,114)(65,95)(66,76)(67,117)(68,98)(69,79)(70,120)(71,101)(72,82)(74,104)(75,85)(77,107)(78,88)(80,110)(81,91)(83,113)(84,94)(86,116)(87,97)(89,119)(90,100)(93,103)(96,106)(99,109)(102,112)(105,115)(108,118)>;
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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71), (1,46,31,16)(2,27,32,57)(3,8,33,38)(4,49,34,19)(5,30,35,60)(6,11,36,41)(7,52,37,22)(9,14,39,44)(10,55,40,25)(12,17,42,47)(13,58,43,28)(15,20,45,50)(18,23,48,53)(21,26,51,56)(24,29,54,59)(61,111)(62,92)(63,73)(64,114)(65,95)(66,76)(67,117)(68,98)(69,79)(70,120)(71,101)(72,82)(74,104)(75,85)(77,107)(78,88)(80,110)(81,91)(83,113)(84,94)(86,116)(87,97)(89,119)(90,100)(93,103)(96,106)(99,109)(102,112)(105,115)(108,118) );
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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,112),(20,111),(21,110),(22,109),(23,108),(24,107),(25,106),(26,105),(27,104),(28,103),(29,102),(30,101),(31,100),(32,99),(33,98),(34,97),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71)], [(1,46,31,16),(2,27,32,57),(3,8,33,38),(4,49,34,19),(5,30,35,60),(6,11,36,41),(7,52,37,22),(9,14,39,44),(10,55,40,25),(12,17,42,47),(13,58,43,28),(15,20,45,50),(18,23,48,53),(21,26,51,56),(24,29,54,59),(61,111),(62,92),(63,73),(64,114),(65,95),(66,76),(67,117),(68,98),(69,79),(70,120),(71,101),(72,82),(74,104),(75,85),(77,107),(78,88),(80,110),(81,91),(83,113),(84,94),(86,116),(87,97),(89,119),(90,100),(93,103),(96,106),(99,109),(102,112),(105,115),(108,118)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 8A | 8B | 10A | ··· | 10F | 12A | 12B | 12C | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 60 | 2 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 60 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | D6 | D10 | C4×S3 | C3⋊D4 | D12 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | D20⋊4C4 | S3×D5 | D12⋊C4 | D30.C2 | C3⋊D20 | C5⋊D12 | D60⋊13C4 |
| kernel | D60⋊13C4 | C3×C4.Dic5 | Dic3×C20 | D60⋊11C2 | Dic30 | D60 | C4.Dic5 | C60 | C2×C30 | C4×Dic3 | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | C15 | C12 | C12 | C2×C6 | C3 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of D60⋊13C4 ►in GL4(𝔽241) generated by
| 6 | 0 | 0 | 0 |
| 0 | 201 | 0 | 0 |
| 0 | 0 | 2 | 44 |
| 0 | 0 | 104 | 240 |
| 0 | 91 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 0 | 67 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 137 | 1 |
G:=sub<GL(4,GF(241))| [6,0,0,0,0,201,0,0,0,0,2,104,0,0,44,240],[0,98,0,0,91,0,0,0,0,0,0,67,0,0,18,0],[64,0,0,0,0,240,0,0,0,0,240,137,0,0,0,1] >;
D60⋊13C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{13}C_4 % in TeX
G:=Group("D60:13C4"); // GroupNames label
G:=SmallGroup(480,56);
// by ID
G=gap.SmallGroup(480,56);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^55*b>;
// generators/relations