metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.29M4(2), C23.(C5:C8), (C2xC20).1C8, C5:2C8.21D4, C5:2(C23.C8), C20.C8:6C2, (C22xC4).4F5, (C22xC10).4C8, (C22xC20).14C4, C4.43(C22:F5), C20.41(C22:C4), C10.10(C22:C8), C4.7(C22.F5), C2.3(C23.2F5), (C2xC4).(C5:C8), C22.2(C2xC5:C8), (C2xC5:2C8).6C4, (C2xC10).27(C2xC8), (C2xC4).129(C2xF5), (C2xC20).144(C2xC4), (C2xC4.Dic5).29C2, (C2xC5:2C8).214C22, SmallGroup(320,250)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.29M4(2)
G = < a,b,c | a20=c2=1, b8=a10, bab-1=a3, ac=ca, cbc=a5b5 >
Subgroups: 162 in 58 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, C23, C10, C10, C16, C2xC8, M4(2), C22xC4, C20, C20, C2xC10, C2xC10, M5(2), C2xM4(2), C5:2C8, C5:2C8, C2xC20, C2xC20, C22xC10, C23.C8, C5:C16, C2xC5:2C8, C4.Dic5, C22xC20, C20.C8, C2xC4.Dic5, C20.29M4(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C22:C4, C2xC8, M4(2), F5, C22:C8, C5:C8, C2xF5, C23.C8, C2xC5:C8, C22.F5, C22:F5, C23.2F5, C20.29M4(2)
►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 80 23 60 6 75 28 55 11 70 33 50 16 65 38 45)(2 67 32 43 7 62 37 58 12 77 22 53 17 72 27 48)(3 74 21 46 8 69 26 41 13 64 31 56 18 79 36 51)(4 61 30 49 9 76 35 44 14 71 40 59 19 66 25 54)(5 68 39 52 10 63 24 47 15 78 29 42 20 73 34 57)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
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), (1,80,23,60,6,75,28,55,11,70,33,50,16,65,38,45)(2,67,32,43,7,62,37,58,12,77,22,53,17,72,27,48)(3,74,21,46,8,69,26,41,13,64,31,56,18,79,36,51)(4,61,30,49,9,76,35,44,14,71,40,59,19,66,25,54)(5,68,39,52,10,63,24,47,15,78,29,42,20,73,34,57), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)>;
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), (1,80,23,60,6,75,28,55,11,70,33,50,16,65,38,45)(2,67,32,43,7,62,37,58,12,77,22,53,17,72,27,48)(3,74,21,46,8,69,26,41,13,64,31,56,18,79,36,51)(4,61,30,49,9,76,35,44,14,71,40,59,19,66,25,54)(5,68,39,52,10,63,24,47,15,78,29,42,20,73,34,57), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80) );
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)], [(1,80,23,60,6,75,28,55,11,70,33,50,16,65,38,45),(2,67,32,43,7,62,37,58,12,77,22,53,17,72,27,48),(3,74,21,46,8,69,26,41,13,64,31,56,18,79,36,51),(4,61,30,49,9,76,35,44,14,71,40,59,19,66,25,54),(5,68,39,52,10,63,24,47,15,78,29,42,20,73,34,57)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10G | 16A | ··· | 16H | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | M4(2) | F5 | C5:C8 | C2xF5 | C5:C8 | C23.C8 | C22.F5 | C22:F5 | C20.29M4(2) |
| kernel | C20.29M4(2) | C20.C8 | C2xC4.Dic5 | C2xC5:2C8 | C22xC20 | C2xC20 | C22xC10 | C5:2C8 | C20 | C22xC4 | C2xC4 | C2xC4 | C23 | C5 | C4 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 |
Matrix representation of C20.29M4(2) ►in GL4(F241) generated by
| 216 | 0 | 0 | 0 |
| 54 | 135 | 0 | 0 |
| 3 | 0 | 6 | 0 |
| 16 | 0 | 0 | 40 |
| 62 | 0 | 239 | 0 |
| 224 | 0 | 79 | 240 |
| 154 | 1 | 179 | 0 |
| 127 | 0 | 197 | 0 |
| 1 | 0 | 0 | 0 |
| 162 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 44 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [216,54,3,16,0,135,0,0,0,0,6,0,0,0,0,40],[62,224,154,127,0,0,1,0,239,79,179,197,0,240,0,0],[1,162,0,44,0,240,0,0,0,0,1,0,0,0,0,240] >;
C20.29M4(2) in GAP, Magma, Sage, TeX
C_{20}._{29}M_4(2) % in TeX
G:=Group("C20.29M4(2)"); // GroupNames label
G:=SmallGroup(320,250);
// by ID
G=gap.SmallGroup(320,250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^20=c^2=1,b^8=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c=a^5*b^5>;
// generators/relations