metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.12C42, D10.13C42, Dic5.11C42, (C2×C8)⋊9F5, (C8×F5)⋊6C2, C4⋊F5.2C4, C4.F5⋊5C4, (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, (C4×D5).87C23, (C8×D5).66C22, (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
Subgroups: 394 in 130 conjugacy classes, 66 normal (40 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×7], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C8○2M4(2), C8×D5 [×4], C2×C5⋊2C8, C2×C40, D5⋊C8 [×2], C4.F5 [×2], C4×F5 [×2], C4⋊F5 [×2], C22.F5 [×2], C22⋊F5 [×2], C2×C4×D5, C8×F5 [×2], C8⋊F5 [×2], D5×C2×C8, D5⋊M4(2), D10.C23, C20.12C42
Quotients:
C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], F5, C2×C42, C8○D4 [×2], C2×F5 [×3], C8○2M4(2), C4×F5 [×2], C22×F5, C2×C4×F5, C20.12C42
Generators and relations
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a3, ac=ca, bc=cb >
►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 42 35 60)(22 49 24 43)(23 56 33 46)(25 50 31 52)(26 57 40 55)(27 44 29 58)(28 51 38 41)(30 45 36 47)(32 59 34 53)(37 54 39 48)
(1 38 79 51 11 28 69 41)(2 39 80 52 12 29 70 42)(3 40 61 53 13 30 71 43)(4 21 62 54 14 31 72 44)(5 22 63 55 15 32 73 45)(6 23 64 56 16 33 74 46)(7 24 65 57 17 34 75 47)(8 25 66 58 18 35 76 48)(9 26 67 59 19 36 77 49)(10 27 68 60 20 37 78 50)
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)(21,42,35,60)(22,49,24,43)(23,56,33,46)(25,50,31,52)(26,57,40,55)(27,44,29,58)(28,51,38,41)(30,45,36,47)(32,59,34,53)(37,54,39,48), (1,38,79,51,11,28,69,41)(2,39,80,52,12,29,70,42)(3,40,61,53,13,30,71,43)(4,21,62,54,14,31,72,44)(5,22,63,55,15,32,73,45)(6,23,64,56,16,33,74,46)(7,24,65,57,17,34,75,47)(8,25,66,58,18,35,76,48)(9,26,67,59,19,36,77,49)(10,27,68,60,20,37,78,50)>;
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)(21,42,35,60)(22,49,24,43)(23,56,33,46)(25,50,31,52)(26,57,40,55)(27,44,29,58)(28,51,38,41)(30,45,36,47)(32,59,34,53)(37,54,39,48), (1,38,79,51,11,28,69,41)(2,39,80,52,12,29,70,42)(3,40,61,53,13,30,71,43)(4,21,62,54,14,31,72,44)(5,22,63,55,15,32,73,45)(6,23,64,56,16,33,74,46)(7,24,65,57,17,34,75,47)(8,25,66,58,18,35,76,48)(9,26,67,59,19,36,77,49)(10,27,68,60,20,37,78,50) );
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),(21,42,35,60),(22,49,24,43),(23,56,33,46),(25,50,31,52),(26,57,40,55),(27,44,29,58),(28,51,38,41),(30,45,36,47),(32,59,34,53),(37,54,39,48)], [(1,38,79,51,11,28,69,41),(2,39,80,52,12,29,70,42),(3,40,61,53,13,30,71,43),(4,21,62,54,14,31,72,44),(5,22,63,55,15,32,73,45),(6,23,64,56,16,33,74,46),(7,24,65,57,17,34,75,47),(8,25,66,58,18,35,76,48),(9,26,67,59,19,36,77,49),(10,27,68,60,20,37,78,50)])
Matrix representation ►G ⊆ 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] >;
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 |
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