metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.36D4, D12⋊21D10, D20.35D6, Dic6⋊18D10, C60.151C23, C4○D12⋊3D5, (C2×D20)⋊8S3, (C6×D20)⋊1C2, C15⋊D8⋊13C2, C15⋊7(C8⋊C22), (C2×C20).85D6, (C2×C30).41D4, C30.73(C2×D4), C60.7C4⋊6C2, C3⋊4(D4⋊D10), (C2×C12).86D10, C5⋊4(D12⋊6C22), C15⋊3C8⋊21C22, C30.D4⋊13C2, (C5×D12)⋊28C22, C4.23(C15⋊D4), C12.26(C5⋊D4), C20.81(C3⋊D4), (C2×C60).31C22, C20.86(C22×S3), C12.86(C22×D5), (C5×Dic6)⋊24C22, (C3×D20).41C22, C22.4(C15⋊D4), (C2×C4).7(S3×D5), C4.124(C2×S3×D5), (C5×C4○D12)⋊1C2, C6.73(C2×C5⋊D4), C2.7(C2×C15⋊D4), C10.74(C2×C3⋊D4), (C2×C6).52(C5⋊D4), (C2×C10).9(C3⋊D4), SmallGroup(480,374)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.36D4
G = < a,b,c | a60=c2=1, b4=a30, bab-1=a-1, cac=a19, cbc=a30b3 >
Subgroups: 668 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×C6, C5×S3, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, D20, D20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C60, C6×D5, S3×C10, C2×C30, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, D12⋊6C22, C15⋊3C8, C3×D20, C3×D20, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D5×C2×C6, D4⋊D10, C15⋊D8, C30.D4, C60.7C4, C6×D20, C5×C4○D12, C60.36D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8⋊C22, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, D12⋊6C22, C15⋊D4, C2×S3×D5, D4⋊D10, C2×C15⋊D4, C60.36D4
►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)
(1 106 16 91 31 76 46 61)(2 105 17 90 32 75 47 120)(3 104 18 89 33 74 48 119)(4 103 19 88 34 73 49 118)(5 102 20 87 35 72 50 117)(6 101 21 86 36 71 51 116)(7 100 22 85 37 70 52 115)(8 99 23 84 38 69 53 114)(9 98 24 83 39 68 54 113)(10 97 25 82 40 67 55 112)(11 96 26 81 41 66 56 111)(12 95 27 80 42 65 57 110)(13 94 28 79 43 64 58 109)(14 93 29 78 44 63 59 108)(15 92 30 77 45 62 60 107)
(2 20)(3 39)(4 58)(5 17)(6 36)(7 55)(8 14)(9 33)(10 52)(12 30)(13 49)(15 27)(16 46)(18 24)(19 43)(22 40)(23 59)(25 37)(26 56)(28 34)(29 53)(32 50)(35 47)(38 44)(42 60)(45 57)(48 54)(61 106)(62 65)(63 84)(64 103)(66 81)(67 100)(68 119)(69 78)(70 97)(71 116)(72 75)(73 94)(74 113)(76 91)(77 110)(79 88)(80 107)(82 85)(83 104)(86 101)(87 120)(89 98)(90 117)(92 95)(93 114)(96 111)(99 108)(102 105)(109 118)(112 115)
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), (1,106,16,91,31,76,46,61)(2,105,17,90,32,75,47,120)(3,104,18,89,33,74,48,119)(4,103,19,88,34,73,49,118)(5,102,20,87,35,72,50,117)(6,101,21,86,36,71,51,116)(7,100,22,85,37,70,52,115)(8,99,23,84,38,69,53,114)(9,98,24,83,39,68,54,113)(10,97,25,82,40,67,55,112)(11,96,26,81,41,66,56,111)(12,95,27,80,42,65,57,110)(13,94,28,79,43,64,58,109)(14,93,29,78,44,63,59,108)(15,92,30,77,45,62,60,107), (2,20)(3,39)(4,58)(5,17)(6,36)(7,55)(8,14)(9,33)(10,52)(12,30)(13,49)(15,27)(16,46)(18,24)(19,43)(22,40)(23,59)(25,37)(26,56)(28,34)(29,53)(32,50)(35,47)(38,44)(42,60)(45,57)(48,54)(61,106)(62,65)(63,84)(64,103)(66,81)(67,100)(68,119)(69,78)(70,97)(71,116)(72,75)(73,94)(74,113)(76,91)(77,110)(79,88)(80,107)(82,85)(83,104)(86,101)(87,120)(89,98)(90,117)(92,95)(93,114)(96,111)(99,108)(102,105)(109,118)(112,115)>;
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), (1,106,16,91,31,76,46,61)(2,105,17,90,32,75,47,120)(3,104,18,89,33,74,48,119)(4,103,19,88,34,73,49,118)(5,102,20,87,35,72,50,117)(6,101,21,86,36,71,51,116)(7,100,22,85,37,70,52,115)(8,99,23,84,38,69,53,114)(9,98,24,83,39,68,54,113)(10,97,25,82,40,67,55,112)(11,96,26,81,41,66,56,111)(12,95,27,80,42,65,57,110)(13,94,28,79,43,64,58,109)(14,93,29,78,44,63,59,108)(15,92,30,77,45,62,60,107), (2,20)(3,39)(4,58)(5,17)(6,36)(7,55)(8,14)(9,33)(10,52)(12,30)(13,49)(15,27)(16,46)(18,24)(19,43)(22,40)(23,59)(25,37)(26,56)(28,34)(29,53)(32,50)(35,47)(38,44)(42,60)(45,57)(48,54)(61,106)(62,65)(63,84)(64,103)(66,81)(67,100)(68,119)(69,78)(70,97)(71,116)(72,75)(73,94)(74,113)(76,91)(77,110)(79,88)(80,107)(82,85)(83,104)(86,101)(87,120)(89,98)(90,117)(92,95)(93,114)(96,111)(99,108)(102,105)(109,118)(112,115) );
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)], [(1,106,16,91,31,76,46,61),(2,105,17,90,32,75,47,120),(3,104,18,89,33,74,48,119),(4,103,19,88,34,73,49,118),(5,102,20,87,35,72,50,117),(6,101,21,86,36,71,51,116),(7,100,22,85,37,70,52,115),(8,99,23,84,38,69,53,114),(9,98,24,83,39,68,54,113),(10,97,25,82,40,67,55,112),(11,96,26,81,41,66,56,111),(12,95,27,80,42,65,57,110),(13,94,28,79,43,64,58,109),(14,93,29,78,44,63,59,108),(15,92,30,77,45,62,60,107)], [(2,20),(3,39),(4,58),(5,17),(6,36),(7,55),(8,14),(9,33),(10,52),(12,30),(13,49),(15,27),(16,46),(18,24),(19,43),(22,40),(23,59),(25,37),(26,56),(28,34),(29,53),(32,50),(35,47),(38,44),(42,60),(45,57),(48,54),(61,106),(62,65),(63,84),(64,103),(66,81),(67,100),(68,119),(69,78),(70,97),(71,116),(72,75),(73,94),(74,113),(76,91),(77,110),(79,88),(80,107),(82,85),(83,104),(86,101),(87,120),(89,98),(90,117),(92,95),(93,114),(96,111),(99,108),(102,105),(109,118),(112,115)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 20 | 20 | 2 | 2 | 2 | 12 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 60 | 60 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | C5⋊D4 | C8⋊C22 | S3×D5 | D12⋊6C22 | C15⋊D4 | C2×S3×D5 | C15⋊D4 | D4⋊D10 | C60.36D4 |
| kernel | C60.36D4 | C15⋊D8 | C30.D4 | C60.7C4 | C6×D20 | C5×C4○D12 | C2×D20 | C60 | C2×C30 | C4○D12 | D20 | C2×C20 | Dic6 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.36D4 ►in GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 189 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 78 | 163 | 0 | 0 |
| 0 | 0 | 78 | 197 | 0 | 0 |
| 0 | 0 | 0 | 0 | 78 | 163 |
| 0 | 0 | 0 | 0 | 78 | 197 |
| 30 | 47 | 0 | 0 | 0 | 0 |
| 227 | 211 | 0 | 0 | 0 | 0 |
| 0 | 0 | 78 | 82 | 132 | 9 |
| 0 | 0 | 204 | 163 | 234 | 109 |
| 0 | 0 | 93 | 83 | 163 | 159 |
| 0 | 0 | 6 | 148 | 37 | 78 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 124 | 188 | 44 | 3 |
| 0 | 0 | 5 | 117 | 78 | 197 |
G:=sub<GL(6,GF(241))| [225,189,0,0,0,0,0,15,0,0,0,0,0,0,78,78,0,0,0,0,163,197,0,0,0,0,0,0,78,78,0,0,0,0,163,197],[30,227,0,0,0,0,47,211,0,0,0,0,0,0,78,204,93,6,0,0,82,163,83,148,0,0,132,234,163,37,0,0,9,109,159,78],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,51,124,5,0,0,0,240,188,117,0,0,0,0,44,78,0,0,0,0,3,197] >;
C60.36D4 in GAP, Magma, Sage, TeX
C_{60}._{36}D_4 % in TeX
G:=Group("C60.36D4"); // GroupNames label
G:=SmallGroup(480,374);
// by ID
G=gap.SmallGroup(480,374);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^-1,c*a*c=a^19,c*b*c=a^30*b^3>;
// generators/relations