metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.12C42, D10.13C42, Dic5.11C42, (C2×C8)⋊9F5, (C8×F5)⋊6C2, C4.F5⋊5C4, C4⋊F5.2C4, (C2×C40)⋊10C4, C4.5(C4×F5), (C8×D5)⋊13C4, C8⋊F5⋊8C2, C8.34(C2×F5), C40.41(C2×C4), C22.F5⋊7C4, C22⋊F5.4C4, C22.4(C4×F5), D5.1(C8○D4), C4.50(C22×F5), C5⋊3(C8○2M4(2)), (C2×C10).18C42, C20.90(C22×C4), C10.13(C2×C42), D5⋊C8.18C22, (C8×D5).66C22, (C4×D5).87C23, (C4×F5).17C22, D5⋊M4(2).12C2, D10.33(C22×C4), Dic5.32(C22×C4), D10.C23.12C2, C5⋊C8.1(C2×C4), C2.14(C2×C4×F5), (D5×C2×C8).34C2, (C2×C5⋊2C8)⋊19C4, (C2×F5).3(C2×C4), C5⋊2C8.54(C2×C4), (C4×D5).66(C2×C4), (C2×C4).135(C2×F5), (C2×C20).146(C2×C4), (C2×C4×D5).402C22, (C22×D5).88(C2×C4), (C2×Dic5).126(C2×C4), SmallGroup(320,1056)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.12C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a3, ac=ca, bc=cb >
Subgroups: 394 in 130 conjugacy classes, 66 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C8○2M4(2), C8×D5, C2×C5⋊2C8, C2×C40, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C22.F5, C22⋊F5, C2×C4×D5, C8×F5, C8⋊F5, D5×C2×C8, D5⋊M4(2), D10.C23, C20.12C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, F5, C2×C42, C8○D4, C2×F5, C8○2M4(2), C4×F5, C22×F5, C2×C4×F5, C20.12C42
►On 80 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)
(1 69 11 79)(2 76 20 62)(3 63 9 65)(4 70 18 68)(5 77 7 71)(6 64 16 74)(8 78 14 80)(10 72 12 66)(13 73 19 75)(15 67 17 61)(21 56 23 50)(22 43 32 53)(24 57 30 59)(25 44 39 42)(26 51 28 45)(27 58 37 48)(29 52 35 54)(31 46 33 60)(34 47 40 49)(36 41 38 55)
(1 32 79 43 11 22 69 53)(2 33 80 44 12 23 70 54)(3 34 61 45 13 24 71 55)(4 35 62 46 14 25 72 56)(5 36 63 47 15 26 73 57)(6 37 64 48 16 27 74 58)(7 38 65 49 17 28 75 59)(8 39 66 50 18 29 76 60)(9 40 67 51 19 30 77 41)(10 21 68 52 20 31 78 42)
G:=sub<Sym(80)| (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), (1,69,11,79)(2,76,20,62)(3,63,9,65)(4,70,18,68)(5,77,7,71)(6,64,16,74)(8,78,14,80)(10,72,12,66)(13,73,19,75)(15,67,17,61)(21,56,23,50)(22,43,32,53)(24,57,30,59)(25,44,39,42)(26,51,28,45)(27,58,37,48)(29,52,35,54)(31,46,33,60)(34,47,40,49)(36,41,38,55), (1,32,79,43,11,22,69,53)(2,33,80,44,12,23,70,54)(3,34,61,45,13,24,71,55)(4,35,62,46,14,25,72,56)(5,36,63,47,15,26,73,57)(6,37,64,48,16,27,74,58)(7,38,65,49,17,28,75,59)(8,39,66,50,18,29,76,60)(9,40,67,51,19,30,77,41)(10,21,68,52,20,31,78,42)>;
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), (1,69,11,79)(2,76,20,62)(3,63,9,65)(4,70,18,68)(5,77,7,71)(6,64,16,74)(8,78,14,80)(10,72,12,66)(13,73,19,75)(15,67,17,61)(21,56,23,50)(22,43,32,53)(24,57,30,59)(25,44,39,42)(26,51,28,45)(27,58,37,48)(29,52,35,54)(31,46,33,60)(34,47,40,49)(36,41,38,55), (1,32,79,43,11,22,69,53)(2,33,80,44,12,23,70,54)(3,34,61,45,13,24,71,55)(4,35,62,46,14,25,72,56)(5,36,63,47,15,26,73,57)(6,37,64,48,16,27,74,58)(7,38,65,49,17,28,75,59)(8,39,66,50,18,29,76,60)(9,40,67,51,19,30,77,41)(10,21,68,52,20,31,78,42) );
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)], [(1,69,11,79),(2,76,20,62),(3,63,9,65),(4,70,18,68),(5,77,7,71),(6,64,16,74),(8,78,14,80),(10,72,12,66),(13,73,19,75),(15,67,17,61),(21,56,23,50),(22,43,32,53),(24,57,30,59),(25,44,39,42),(26,51,28,45),(27,58,37,48),(29,52,35,54),(31,46,33,60),(34,47,40,49),(36,41,38,55)], [(1,32,79,43,11,22,69,53),(2,33,80,44,12,23,70,54),(3,34,61,45,13,24,71,55),(4,35,62,46,14,25,72,56),(5,36,63,47,15,26,73,57),(6,37,64,48,16,27,74,58),(7,38,65,49,17,28,75,59),(8,39,66,50,18,29,76,60),(9,40,67,51,19,30,77,41),(10,21,68,52,20,31,78,42)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4N | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | ··· | 8T | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | ··· | 10 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4 | C8○D4 | F5 | C2×F5 | C2×F5 | C4×F5 | C4×F5 | C20.12C42 |
| kernel | C20.12C42 | C8×F5 | C8⋊F5 | D5×C2×C8 | D5⋊M4(2) | D10.C23 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4.F5 | C4⋊F5 | C22.F5 | C22⋊F5 | D5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C20.12C42 ►in GL4(𝔽41) generated by
| 34 | 7 | 7 | 34 |
| 7 | 0 | 14 | 14 |
| 27 | 34 | 27 | 0 |
| 0 | 27 | 34 | 27 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 9 | 0 | 0 |
| 32 | 32 | 32 | 32 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(41))| [34,7,27,0,7,0,34,27,7,14,27,34,34,14,0,27],[9,0,0,32,0,0,9,32,0,0,0,32,0,9,0,32],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27] >;
C20.12C42 in GAP, Magma, Sage, TeX
C_{20}._{12}C_4^2 % in TeX
G:=Group("C20.12C4^2"); // GroupNames label
G:=SmallGroup(320,1056);
// by ID
G=gap.SmallGroup(320,1056);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^3,a*c=c*a,b*c=c*b>;
// generators/relations