metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.32C42, C4⋊C4⋊4Dic5, C10.25C4≀C2, (C2×C20).7Q8, (C4×Dic5)⋊8C4, C4.Dic5⋊9C4, C20.39(C4⋊C4), C4.2(C4×Dic5), C5⋊5(C42⋊6C4), (C2×C4).126D20, (C2×C20).488D4, C4.2(C4⋊Dic5), (C2×C4).24Dic10, (C22×C10).43D4, C42⋊C2.2D5, (C22×C4).324D10, C23.47(C5⋊D4), C4.46(D10⋊C4), C20.108(C22⋊C4), C2.1(D4⋊2Dic5), C4.30(C10.D4), (C22×C20).121C22, C22.4(C10.D4), C22.28(C23.D5), C22.18(D10⋊C4), C10.27(C2.C42), C2.9(C10.10C42), (C5×C4⋊C4)⋊13C4, (C2×C4).68(C4×D5), (C2×C4×Dic5).1C2, (C2×C10).32(C4⋊C4), (C2×C20).230(C2×C4), (C2×C4).36(C2×Dic5), (C2×C4.Dic5).8C2, (C2×C4).268(C5⋊D4), (C5×C42⋊C2).2C2, (C2×C10).154(C22⋊C4), SmallGroup(320,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.32C42
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, cac-1=a11, cbc-1=a5b >
Subgroups: 326 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C22×C10, C42⋊6C4, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×Dic5, C4×Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C2×C4.Dic5, C2×C4×Dic5, C5×C42⋊C2, C20.32C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4≀C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C42⋊6C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, D4⋊2Dic5, C20.32C42
►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 71 31 46 11 61 21 56)(2 70 32 45 12 80 22 55)(3 69 33 44 13 79 23 54)(4 68 34 43 14 78 24 53)(5 67 35 42 15 77 25 52)(6 66 36 41 16 76 26 51)(7 65 37 60 17 75 27 50)(8 64 38 59 18 74 28 49)(9 63 39 58 19 73 29 48)(10 62 40 57 20 72 30 47)
(1 61 26 41)(2 72 27 52)(3 63 28 43)(4 74 29 54)(5 65 30 45)(6 76 31 56)(7 67 32 47)(8 78 33 58)(9 69 34 49)(10 80 35 60)(11 71 36 51)(12 62 37 42)(13 73 38 53)(14 64 39 44)(15 75 40 55)(16 66 21 46)(17 77 22 57)(18 68 23 48)(19 79 24 59)(20 70 25 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), (1,71,31,46,11,61,21,56)(2,70,32,45,12,80,22,55)(3,69,33,44,13,79,23,54)(4,68,34,43,14,78,24,53)(5,67,35,42,15,77,25,52)(6,66,36,41,16,76,26,51)(7,65,37,60,17,75,27,50)(8,64,38,59,18,74,28,49)(9,63,39,58,19,73,29,48)(10,62,40,57,20,72,30,47), (1,61,26,41)(2,72,27,52)(3,63,28,43)(4,74,29,54)(5,65,30,45)(6,76,31,56)(7,67,32,47)(8,78,33,58)(9,69,34,49)(10,80,35,60)(11,71,36,51)(12,62,37,42)(13,73,38,53)(14,64,39,44)(15,75,40,55)(16,66,21,46)(17,77,22,57)(18,68,23,48)(19,79,24,59)(20,70,25,50)>;
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), (1,71,31,46,11,61,21,56)(2,70,32,45,12,80,22,55)(3,69,33,44,13,79,23,54)(4,68,34,43,14,78,24,53)(5,67,35,42,15,77,25,52)(6,66,36,41,16,76,26,51)(7,65,37,60,17,75,27,50)(8,64,38,59,18,74,28,49)(9,63,39,58,19,73,29,48)(10,62,40,57,20,72,30,47), (1,61,26,41)(2,72,27,52)(3,63,28,43)(4,74,29,54)(5,65,30,45)(6,76,31,56)(7,67,32,47)(8,78,33,58)(9,69,34,49)(10,80,35,60)(11,71,36,51)(12,62,37,42)(13,73,38,53)(14,64,39,44)(15,75,40,55)(16,66,21,46)(17,77,22,57)(18,68,23,48)(19,79,24,59)(20,70,25,50) );
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)], [(1,71,31,46,11,61,21,56),(2,70,32,45,12,80,22,55),(3,69,33,44,13,79,23,54),(4,68,34,43,14,78,24,53),(5,67,35,42,15,77,25,52),(6,66,36,41,16,76,26,51),(7,65,37,60,17,75,27,50),(8,64,38,59,18,74,28,49),(9,63,39,58,19,73,29,48),(10,62,40,57,20,72,30,47)], [(1,61,26,41),(2,72,27,52),(3,63,28,43),(4,74,29,54),(5,65,30,45),(6,76,31,56),(7,67,32,47),(8,78,33,58),(9,69,34,49),(10,80,35,60),(11,71,36,51),(12,62,37,42),(13,73,38,53),(14,64,39,44),(15,75,40,55),(16,66,21,46),(17,77,22,57),(18,68,23,48),(19,79,24,59),(20,70,25,50)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4R | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D5 | Dic5 | D10 | C4≀C2 | Dic10 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | D4⋊2Dic5 |
| kernel | C20.32C42 | C2×C4.Dic5 | C2×C4×Dic5 | C5×C42⋊C2 | C4.Dic5 | C4×Dic5 | C5×C4⋊C4 | C2×C20 | C2×C20 | C22×C10 | C42⋊C2 | C4⋊C4 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 4 | 4 | 8 |
Matrix representation of C20.32C42 ►in GL4(𝔽41) generated by
| 9 | 6 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 35 | 1 |
| 0 | 0 | 40 | 0 |
| 15 | 38 | 0 | 0 |
| 37 | 26 | 0 | 0 |
| 0 | 0 | 39 | 28 |
| 0 | 0 | 16 | 2 |
| 26 | 5 | 0 | 0 |
| 4 | 15 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 28 | 39 |
G:=sub<GL(4,GF(41))| [9,0,0,0,6,32,0,0,0,0,35,40,0,0,1,0],[15,37,0,0,38,26,0,0,0,0,39,16,0,0,28,2],[26,4,0,0,5,15,0,0,0,0,2,28,0,0,13,39] >;
C20.32C42 in GAP, Magma, Sage, TeX
C_{20}._{32}C_4^2 % in TeX
G:=Group("C20.32C4^2"); // GroupNames label
G:=SmallGroup(320,90);
// by ID
G=gap.SmallGroup(320,90);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;
// generators/relations