G = C60.59(C2×C4) order 480 = 25·3·5
13rd non-split extension by C60 of C2×C4 acting via C2×C4/C2=C4
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C60).6C4, C60.59(C2×C4), (C4×D5).96D6, D5⋊(C4.Dic3), C12.52(C2×F5), (C2×C12).12F5, C60.C4⋊9C2, (D5×C12).13C4, C3⋊4(D5⋊M4(2)), (C3×D5)⋊4M4(2), C15⋊12(C2×M4(2)), (C4×D5).5Dic3, C12.F5⋊11C2, (C2×C20).8Dic3, C15⋊C8⋊12C22, C6.34(C22×F5), C15⋊8M4(2)⋊7C2, C30.72(C22×C4), C20.13(C2×Dic3), D10.14(C2×Dic3), (C2×Dic5).208D6, (C22×D5).9Dic3, C10.3(C22×Dic3), Dic5.16(C2×Dic3), (D5×C12).124C22, (C3×Dic5).64C23, Dic5.50(C22×S3), (C6×Dic5).267C22, C4.20(C2×C3⋊F5), (C2×C4×D5).16S3, (D5×C2×C6).15C4, (C2×C4).8(C3⋊F5), C5⋊2(C2×C4.Dic3), C2.5(C22×C3⋊F5), C22.6(C2×C3⋊F5), (D5×C2×C12).19C2, (C2×C6).45(C2×F5), (C2×C30).39(C2×C4), (C6×D5).58(C2×C4), (C3×Dic5).66(C2×C4), (C2×C10).15(C2×Dic3), SmallGroup(480,1062)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.59(C2×C4)
G = < a,b,c | a60=b2=1, c4=a30, ab=ba, cac-1=a17, cbc-1=a30b >
Subgroups: 524 in 136 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C4.Dic3, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C2×C4.Dic3, C15⋊C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5⋊M4(2), C60.C4, C12.F5, C15⋊8M4(2), D5×C2×C12, C60.59(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, F5, C2×Dic3, C22×S3, C2×M4(2), C2×F5, C4.Dic3, C22×Dic3, C3⋊F5, C22×F5, C2×C4.Dic3, C2×C3⋊F5, D5⋊M4(2), C22×C3⋊F5, C60.59(C2×C4)
►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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 116 46 101 31 86 16 71)(2 109 35 118 32 79 5 88)(3 102 24 75 33 72 54 105)(4 95 13 92 34 65 43 62)(6 81 51 66 36 111 21 96)(7 74 40 83 37 104 10 113)(8 67 29 100 38 97 59 70)(9 120 18 117 39 90 48 87)(11 106 56 91 41 76 26 61)(12 99 45 108 42 69 15 78)(14 85 23 82 44 115 53 112)(17 64 50 73 47 94 20 103)(19 110 28 107 49 80 58 77)(22 89 55 98 52 119 25 68)(27 114 60 63 57 84 30 93)
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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60), (1,116,46,101,31,86,16,71)(2,109,35,118,32,79,5,88)(3,102,24,75,33,72,54,105)(4,95,13,92,34,65,43,62)(6,81,51,66,36,111,21,96)(7,74,40,83,37,104,10,113)(8,67,29,100,38,97,59,70)(9,120,18,117,39,90,48,87)(11,106,56,91,41,76,26,61)(12,99,45,108,42,69,15,78)(14,85,23,82,44,115,53,112)(17,64,50,73,47,94,20,103)(19,110,28,107,49,80,58,77)(22,89,55,98,52,119,25,68)(27,114,60,63,57,84,30,93)>;
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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60), (1,116,46,101,31,86,16,71)(2,109,35,118,32,79,5,88)(3,102,24,75,33,72,54,105)(4,95,13,92,34,65,43,62)(6,81,51,66,36,111,21,96)(7,74,40,83,37,104,10,113)(8,67,29,100,38,97,59,70)(9,120,18,117,39,90,48,87)(11,106,56,91,41,76,26,61)(12,99,45,108,42,69,15,78)(14,85,23,82,44,115,53,112)(17,64,50,73,47,94,20,103)(19,110,28,107,49,80,58,77)(22,89,55,98,52,119,25,68)(27,114,60,63,57,84,30,93) );
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)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60)], [(1,116,46,101,31,86,16,71),(2,109,35,118,32,79,5,88),(3,102,24,75,33,72,54,105),(4,95,13,92,34,65,43,62),(6,81,51,66,36,111,21,96),(7,74,40,83,37,104,10,113),(8,67,29,100,38,97,59,70),(9,120,18,117,39,90,48,87),(11,106,56,91,41,76,26,61),(12,99,45,108,42,69,15,78),(14,85,23,82,44,115,53,112),(17,64,50,73,47,94,20,103),(19,110,28,107,49,80,58,77),(22,89,55,98,52,119,25,68),(27,114,60,63,57,84,30,93)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 1 | 1 | 2 | 5 | 5 | 10 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | - | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | D6 | Dic3 | Dic3 | M4(2) | C4.Dic3 | F5 | C2×F5 | C2×F5 | C3⋊F5 | C2×C3⋊F5 | C2×C3⋊F5 | D5⋊M4(2) | C60.59(C2×C4) |
| kernel | C60.59(C2×C4) | C60.C4 | C12.F5 | C15⋊8M4(2) | D5×C2×C12 | D5×C12 | C2×C60 | D5×C2×C6 | C2×C4×D5 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C3×D5 | D5 | C2×C12 | C12 | C2×C6 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of C60.59(C2×C4) ►in GL6(𝔽241)
| 11 | 236 | 0 | 0 | 0 | 0 |
| 215 | 231 | 0 | 0 | 0 | 0 |
| 0 | 0 | 110 | 110 | 0 | 0 |
| 0 | 0 | 131 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 131 | 177 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 1 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 92 | 189 | 0 | 0 | 0 | 0 |
| 223 | 149 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 131 | 177 | 0 | 0 |
| 0 | 0 | 110 | 110 | 0 | 0 |
G:=sub<GL(6,GF(241))| [11,215,0,0,0,0,236,231,0,0,0,0,0,0,110,131,0,0,0,0,110,177,0,0,0,0,0,0,131,64,0,0,0,0,177,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[92,223,0,0,0,0,189,149,0,0,0,0,0,0,0,0,131,110,0,0,0,0,177,110,0,0,1,0,0,0,0,0,0,1,0,0] >;
C60.59(C2×C4) in GAP, Magma, Sage, TeX
C_{60}._{59}(C_2\times C_4) % in TeX
G:=Group("C60.59(C2xC4)"); // GroupNames label
G:=SmallGroup(480,1062);
// by ID
G=gap.SmallGroup(480,1062);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,80,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=1,c^4=a^30,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^30*b>;
// generators/relations