?
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), (C2×C12).12F5, C12.52(C2×F5), 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, Dic5.50(C22×S3), (C3×Dic5).64C23, (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
Subgroups: 524 in 136 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C3⋊C8 [×4], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C2×M4(2), C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8 [×2], C4.Dic3 [×4], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×2], C2×C4×D5, C2×C4.Dic3, C15⋊C8 [×4], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, D5⋊M4(2), C60.C4 [×2], C12.F5 [×2], C15⋊8M4(2) [×2], D5×C2×C12, C60.59(C2×C4)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, F5, C2×Dic3 [×6], C22×S3, C2×M4(2), C2×F5 [×3], C4.Dic3 [×2], C22×Dic3, C3⋊F5, C22×F5, C2×C4.Dic3, C2×C3⋊F5 [×3], D5⋊M4(2), C22×C3⋊F5, C60.59(C2×C4)
Generators and relations
G = < a,b,c | a60=b2=1, c4=a30, ab=ba, cac-1=a17, cbc-1=a30b >
►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 61 46 106 31 91 16 76)(2 114 35 63 32 84 5 93)(3 107 24 80 33 77 54 110)(4 100 13 97 34 70 43 67)(6 86 51 71 36 116 21 101)(7 79 40 88 37 109 10 118)(8 72 29 105 38 102 59 75)(9 65 18 62 39 95 48 92)(11 111 56 96 41 81 26 66)(12 104 45 113 42 74 15 83)(14 90 23 87 44 120 53 117)(17 69 50 78 47 99 20 108)(19 115 28 112 49 85 58 82)(22 94 55 103 52 64 25 73)(27 119 60 68 57 89 30 98)
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), (1,61,46,106,31,91,16,76)(2,114,35,63,32,84,5,93)(3,107,24,80,33,77,54,110)(4,100,13,97,34,70,43,67)(6,86,51,71,36,116,21,101)(7,79,40,88,37,109,10,118)(8,72,29,105,38,102,59,75)(9,65,18,62,39,95,48,92)(11,111,56,96,41,81,26,66)(12,104,45,113,42,74,15,83)(14,90,23,87,44,120,53,117)(17,69,50,78,47,99,20,108)(19,115,28,112,49,85,58,82)(22,94,55,103,52,64,25,73)(27,119,60,68,57,89,30,98)>;
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), (1,61,46,106,31,91,16,76)(2,114,35,63,32,84,5,93)(3,107,24,80,33,77,54,110)(4,100,13,97,34,70,43,67)(6,86,51,71,36,116,21,101)(7,79,40,88,37,109,10,118)(8,72,29,105,38,102,59,75)(9,65,18,62,39,95,48,92)(11,111,56,96,41,81,26,66)(12,104,45,113,42,74,15,83)(14,90,23,87,44,120,53,117)(17,69,50,78,47,99,20,108)(19,115,28,112,49,85,58,82)(22,94,55,103,52,64,25,73)(27,119,60,68,57,89,30,98) );
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)], [(1,61,46,106,31,91,16,76),(2,114,35,63,32,84,5,93),(3,107,24,80,33,77,54,110),(4,100,13,97,34,70,43,67),(6,86,51,71,36,116,21,101),(7,79,40,88,37,109,10,118),(8,72,29,105,38,102,59,75),(9,65,18,62,39,95,48,92),(11,111,56,96,41,81,26,66),(12,104,45,113,42,74,15,83),(14,90,23,87,44,120,53,117),(17,69,50,78,47,99,20,108),(19,115,28,112,49,85,58,82),(22,94,55,103,52,64,25,73),(27,119,60,68,57,89,30,98)])
Matrix representation ►G ⊆ 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] >;
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 |
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