metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.96D4, D20⋊7Dic3, Dic10⋊7Dic3, C15⋊5C4≀C2, (C2×C6).1D20, C12.7(C4×D5), (C3×D20)⋊11C4, C4○D20.2S3, (C2×C20).54D6, (C2×C30).23D4, C4.Dic3⋊4D5, C4.8(D5×Dic3), C3⋊3(D20⋊7C4), C60.124(C2×C4), (C2×C12).55D10, (C3×Dic10)⋊11C4, (C4×Dic15)⋊28C2, C5⋊3(Q8⋊3Dic3), C4.28(C15⋊D4), C12.86(C5⋊D4), C20.86(C3⋊D4), C20.24(C2×Dic3), C30.59(C22⋊C4), (C2×C60).210C22, C22.7(C3⋊D20), C6.30(D10⋊C4), C2.9(D10⋊Dic3), C10.19(C6.D4), (C3×C4○D20).5C2, (C2×C4).188(S3×D5), (C2×C10).1(C3⋊D4), (C5×C4.Dic3)⋊10C2, SmallGroup(480,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.96D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a49, cbc=a45b-1 >
Subgroups: 428 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C4×Dic5, C5×M4(2), C4○D20, Q8⋊3Dic3, C5×C3⋊C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×Dic15, C2×C60, D20⋊7C4, C5×C4.Dic3, C4×Dic15, C3×C4○D20, C60.96D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, Q8⋊3Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D20⋊7C4, D10⋊Dic3, C60.96D4
►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 46 31 16)(2 15 32 45)(3 44 33 14)(4 13 34 43)(5 42 35 12)(6 11 36 41)(7 40 37 10)(8 9 38 39)(17 30 47 60)(18 59 48 29)(19 28 49 58)(20 57 50 27)(21 26 51 56)(22 55 52 25)(23 24 53 54)(61 73)(62 102)(63 71)(64 100)(65 69)(66 98)(68 96)(70 94)(72 92)(74 90)(75 119)(76 88)(77 117)(78 86)(79 115)(80 84)(81 113)(83 111)(85 109)(87 107)(89 105)(91 103)(93 101)(95 99)(104 120)(106 118)(108 116)(110 114)
(1 67)(2 116)(3 105)(4 94)(5 83)(6 72)(7 61)(8 110)(9 99)(10 88)(11 77)(12 66)(13 115)(14 104)(15 93)(16 82)(17 71)(18 120)(19 109)(20 98)(21 87)(22 76)(23 65)(24 114)(25 103)(26 92)(27 81)(28 70)(29 119)(30 108)(31 97)(32 86)(33 75)(34 64)(35 113)(36 102)(37 91)(38 80)(39 69)(40 118)(41 107)(42 96)(43 85)(44 74)(45 63)(46 112)(47 101)(48 90)(49 79)(50 68)(51 117)(52 106)(53 95)(54 84)(55 73)(56 62)(57 111)(58 100)(59 89)(60 78)
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), (1,46,31,16)(2,15,32,45)(3,44,33,14)(4,13,34,43)(5,42,35,12)(6,11,36,41)(7,40,37,10)(8,9,38,39)(17,30,47,60)(18,59,48,29)(19,28,49,58)(20,57,50,27)(21,26,51,56)(22,55,52,25)(23,24,53,54)(61,73)(62,102)(63,71)(64,100)(65,69)(66,98)(68,96)(70,94)(72,92)(74,90)(75,119)(76,88)(77,117)(78,86)(79,115)(80,84)(81,113)(83,111)(85,109)(87,107)(89,105)(91,103)(93,101)(95,99)(104,120)(106,118)(108,116)(110,114), (1,67)(2,116)(3,105)(4,94)(5,83)(6,72)(7,61)(8,110)(9,99)(10,88)(11,77)(12,66)(13,115)(14,104)(15,93)(16,82)(17,71)(18,120)(19,109)(20,98)(21,87)(22,76)(23,65)(24,114)(25,103)(26,92)(27,81)(28,70)(29,119)(30,108)(31,97)(32,86)(33,75)(34,64)(35,113)(36,102)(37,91)(38,80)(39,69)(40,118)(41,107)(42,96)(43,85)(44,74)(45,63)(46,112)(47,101)(48,90)(49,79)(50,68)(51,117)(52,106)(53,95)(54,84)(55,73)(56,62)(57,111)(58,100)(59,89)(60,78)>;
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), (1,46,31,16)(2,15,32,45)(3,44,33,14)(4,13,34,43)(5,42,35,12)(6,11,36,41)(7,40,37,10)(8,9,38,39)(17,30,47,60)(18,59,48,29)(19,28,49,58)(20,57,50,27)(21,26,51,56)(22,55,52,25)(23,24,53,54)(61,73)(62,102)(63,71)(64,100)(65,69)(66,98)(68,96)(70,94)(72,92)(74,90)(75,119)(76,88)(77,117)(78,86)(79,115)(80,84)(81,113)(83,111)(85,109)(87,107)(89,105)(91,103)(93,101)(95,99)(104,120)(106,118)(108,116)(110,114), (1,67)(2,116)(3,105)(4,94)(5,83)(6,72)(7,61)(8,110)(9,99)(10,88)(11,77)(12,66)(13,115)(14,104)(15,93)(16,82)(17,71)(18,120)(19,109)(20,98)(21,87)(22,76)(23,65)(24,114)(25,103)(26,92)(27,81)(28,70)(29,119)(30,108)(31,97)(32,86)(33,75)(34,64)(35,113)(36,102)(37,91)(38,80)(39,69)(40,118)(41,107)(42,96)(43,85)(44,74)(45,63)(46,112)(47,101)(48,90)(49,79)(50,68)(51,117)(52,106)(53,95)(54,84)(55,73)(56,62)(57,111)(58,100)(59,89)(60,78) );
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)], [(1,46,31,16),(2,15,32,45),(3,44,33,14),(4,13,34,43),(5,42,35,12),(6,11,36,41),(7,40,37,10),(8,9,38,39),(17,30,47,60),(18,59,48,29),(19,28,49,58),(20,57,50,27),(21,26,51,56),(22,55,52,25),(23,24,53,54),(61,73),(62,102),(63,71),(64,100),(65,69),(66,98),(68,96),(70,94),(72,92),(74,90),(75,119),(76,88),(77,117),(78,86),(79,115),(80,84),(81,113),(83,111),(85,109),(87,107),(89,105),(91,103),(93,101),(95,99),(104,120),(106,118),(108,116),(110,114)], [(1,67),(2,116),(3,105),(4,94),(5,83),(6,72),(7,61),(8,110),(9,99),(10,88),(11,77),(12,66),(13,115),(14,104),(15,93),(16,82),(17,71),(18,120),(19,109),(20,98),(21,87),(22,76),(23,65),(24,114),(25,103),(26,92),(27,81),(28,70),(29,119),(30,108),(31,97),(32,86),(33,75),(34,64),(35,113),(36,102),(37,91),(38,80),(39,69),(40,118),(41,107),(42,96),(43,85),(44,74),(45,63),(46,112),(47,101),(48,90),(49,79),(50,68),(51,117),(52,106),(53,95),(54,84),(55,73),(56,62),(57,111),(58,100),(59,89),(60,78)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 2 | 1 | 1 | 2 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 12 | 12 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
60 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 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | Dic3 | Dic3 | D6 | D10 | C3⋊D4 | C3⋊D4 | C4≀C2 | C4×D5 | C5⋊D4 | D20 | S3×D5 | Q8⋊3Dic3 | D5×Dic3 | C15⋊D4 | C3⋊D20 | D20⋊7C4 | C60.96D4 |
| kernel | C60.96D4 | C5×C4.Dic3 | C4×Dic15 | C3×C4○D20 | C3×Dic10 | C3×D20 | C4○D20 | C60 | C2×C30 | C4.Dic3 | Dic10 | D20 | C2×C20 | C2×C12 | C20 | C2×C10 | C15 | C12 | C12 | C2×C6 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.96D4 ►in GL6(𝔽241)
| 52 | 52 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 54 | 0 | 0 |
| 0 | 0 | 64 | 207 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 54 | 0 | 0 |
| 0 | 0 | 233 | 207 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 177 |
| 0 | 0 | 0 | 0 | 64 | 0 |
G:=sub<GL(6,GF(241))| [52,189,0,0,0,0,52,240,0,0,0,0,0,0,33,64,0,0,0,0,54,207,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[1,189,0,0,0,0,0,240,0,0,0,0,0,0,34,233,0,0,0,0,54,207,0,0,0,0,0,0,64,0,0,0,0,0,0,1],[240,52,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,64,0,0,0,0,177,0] >;
C60.96D4 in GAP, Magma, Sage, TeX
C_{60}._{96}D_4 % in TeX
G:=Group("C60.96D4"); // GroupNames label
G:=SmallGroup(480,52);
// by ID
G=gap.SmallGroup(480,52);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^49,c*b*c=a^45*b^-1>;
// generators/relations