metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.51C42, C23.3Dic10, (C2×C40).27C4, (C2×C4).15D20, (C2×C8).3Dic5, (C2×C20).113D4, C4.26(C4×Dic5), (C22×C10).9Q8, (C22×C4).65D10, C5⋊2(C4.10C42), C20.59(C22⋊C4), (C2×M4(2)).11D5, C4.20(C23.D5), C4.36(D10⋊C4), (C10×M4(2)).15C2, C22.11(C4⋊Dic5), (C22×C20).129C22, C22.7(C10.D4), C10.38(C2.C42), C2.19(C10.10C42), (C2×C5⋊2C8).3C4, (C2×C4).143(C4×D5), (C2×C10).38(C4⋊C4), (C2×C20).239(C2×C4), (C2×C4).23(C5⋊D4), (C2×C4).78(C2×Dic5), (C2×C4.Dic5).14C2, SmallGroup(320,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.51C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a9, ac=ca, cbc-1=a5b >
Subgroups: 214 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×M4(2), C5⋊2C8, C40, C2×C20, C2×C20, C22×C10, C4.10C42, C2×C5⋊2C8, C4.Dic5, C2×C40, C5×M4(2), C22×C20, C2×C4.Dic5, C10×M4(2), C20.51C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4.10C42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, C20.51C42
►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 45 16 60 11 55 6 50)(2 54 17 49 12 44 7 59)(3 43 18 58 13 53 8 48)(4 52 19 47 14 42 9 57)(5 41 20 56 15 51 10 46)(21 63 26 68 31 73 36 78)(22 72 27 77 32 62 37 67)(23 61 28 66 33 71 38 76)(24 70 29 75 34 80 39 65)(25 79 30 64 35 69 40 74)
(1 31 16 26 11 21 6 36)(2 32 17 27 12 22 7 37)(3 33 18 28 13 23 8 38)(4 34 19 29 14 24 9 39)(5 35 20 30 15 25 10 40)(41 64 46 69 51 74 56 79)(42 65 47 70 52 75 57 80)(43 66 48 71 53 76 58 61)(44 67 49 72 54 77 59 62)(45 68 50 73 55 78 60 63)
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), (1,45,16,60,11,55,6,50)(2,54,17,49,12,44,7,59)(3,43,18,58,13,53,8,48)(4,52,19,47,14,42,9,57)(5,41,20,56,15,51,10,46)(21,63,26,68,31,73,36,78)(22,72,27,77,32,62,37,67)(23,61,28,66,33,71,38,76)(24,70,29,75,34,80,39,65)(25,79,30,64,35,69,40,74), (1,31,16,26,11,21,6,36)(2,32,17,27,12,22,7,37)(3,33,18,28,13,23,8,38)(4,34,19,29,14,24,9,39)(5,35,20,30,15,25,10,40)(41,64,46,69,51,74,56,79)(42,65,47,70,52,75,57,80)(43,66,48,71,53,76,58,61)(44,67,49,72,54,77,59,62)(45,68,50,73,55,78,60,63)>;
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), (1,45,16,60,11,55,6,50)(2,54,17,49,12,44,7,59)(3,43,18,58,13,53,8,48)(4,52,19,47,14,42,9,57)(5,41,20,56,15,51,10,46)(21,63,26,68,31,73,36,78)(22,72,27,77,32,62,37,67)(23,61,28,66,33,71,38,76)(24,70,29,75,34,80,39,65)(25,79,30,64,35,69,40,74), (1,31,16,26,11,21,6,36)(2,32,17,27,12,22,7,37)(3,33,18,28,13,23,8,38)(4,34,19,29,14,24,9,39)(5,35,20,30,15,25,10,40)(41,64,46,69,51,74,56,79)(42,65,47,70,52,75,57,80)(43,66,48,71,53,76,58,61)(44,67,49,72,54,77,59,62)(45,68,50,73,55,78,60,63) );
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)], [(1,45,16,60,11,55,6,50),(2,54,17,49,12,44,7,59),(3,43,18,58,13,53,8,48),(4,52,19,47,14,42,9,57),(5,41,20,56,15,51,10,46),(21,63,26,68,31,73,36,78),(22,72,27,77,32,62,37,67),(23,61,28,66,33,71,38,76),(24,70,29,75,34,80,39,65),(25,79,30,64,35,69,40,74)], [(1,31,16,26,11,21,6,36),(2,32,17,27,12,22,7,37),(3,33,18,28,13,23,8,38),(4,34,19,29,14,24,9,39),(5,35,20,30,15,25,10,40),(41,64,46,69,51,74,56,79),(42,65,47,70,52,75,57,80),(43,66,48,71,53,76,58,61),(44,67,49,72,54,77,59,62),(45,68,50,73,55,78,60,63)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | Dic5 | D10 | C4×D5 | D20 | C5⋊D4 | Dic10 | C4.10C42 | C20.51C42 |
| kernel | C20.51C42 | C2×C4.Dic5 | C10×M4(2) | C2×C5⋊2C8 | C2×C40 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 2 | 8 |
Matrix representation of C20.51C42 ►in GL6(𝔽41)
| 40 | 40 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 26 | 3 | 0 | 0 | 0 | 0 |
| 21 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 9 | 0 |
| 0 | 0 | 11 | 0 | 20 | 1 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 32 | 9 | 31 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 9 | 0 | 0 |
| 0 | 0 | 32 | 20 | 0 | 0 |
| 0 | 0 | 29 | 14 | 0 | 9 |
| 0 | 0 | 7 | 31 | 1 | 0 |
G:=sub<GL(6,GF(41))| [40,8,0,0,0,0,40,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[26,21,0,0,0,0,3,15,0,0,0,0,0,0,27,11,0,32,0,0,0,0,0,9,0,0,9,20,14,31,0,0,0,1,0,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,32,29,7,0,0,9,20,14,31,0,0,0,0,0,1,0,0,0,0,9,0] >;
C20.51C42 in GAP, Magma, Sage, TeX
C_{20}._{51}C_4^2 % in TeX
G:=Group("C20.51C4^2"); // GroupNames label
G:=SmallGroup(320,118);
// by ID
G=gap.SmallGroup(320,118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1123,136,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^9,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations