metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20:8C4, Dic5:5D4, C5:4(C4xD4), C4:C4:8D5, C4:1(C4xD5), C20:5(C2xC4), C2.4(D4xD5), D10:4(C2xC4), (C4xDic5):3C2, (C2xD20).8C2, (C2xC4).31D10, C10.24(C2xD4), D10:C4:12C2, C10.33(C4oD4), (C2xC10).34C23, (C2xC20).24C22, C10.24(C22xC4), C2.2(Q8:2D5), C22.18(C22xD5), (C2xDic5).63C22, (C22xD5).23C22, (C5xC4:C4):4C2, (C2xC4xD5):12C2, C2.13(C2xC4xD5), SmallGroup(160,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20:8C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >
Subgroups: 312 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, C23, D5, C10, C42, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C4xD4, C4xD5, D20, C2xDic5, C2xC20, C2xC20, C22xD5, C4xDic5, D10:C4, C5xC4:C4, C2xC4xD5, C2xD20, D20:8C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22xC4, C2xD4, C4oD4, D10, C4xD4, C4xD5, C22xD5, C2xC4xD5, D4xD5, Q8:2D5, D20:8C4
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)
(1 77 55 29)(2 68 56 40)(3 79 57 31)(4 70 58 22)(5 61 59 33)(6 72 60 24)(7 63 41 35)(8 74 42 26)(9 65 43 37)(10 76 44 28)(11 67 45 39)(12 78 46 30)(13 69 47 21)(14 80 48 32)(15 71 49 23)(16 62 50 34)(17 73 51 25)(18 64 52 36)(19 75 53 27)(20 66 54 38)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,43)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80), (1,77,55,29)(2,68,56,40)(3,79,57,31)(4,70,58,22)(5,61,59,33)(6,72,60,24)(7,63,41,35)(8,74,42,26)(9,65,43,37)(10,76,44,28)(11,67,45,39)(12,78,46,30)(13,69,47,21)(14,80,48,32)(15,71,49,23)(16,62,50,34)(17,73,51,25)(18,64,52,36)(19,75,53,27)(20,66,54,38)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,43)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80), (1,77,55,29)(2,68,56,40)(3,79,57,31)(4,70,58,22)(5,61,59,33)(6,72,60,24)(7,63,41,35)(8,74,42,26)(9,65,43,37)(10,76,44,28)(11,67,45,39)(12,78,46,30)(13,69,47,21)(14,80,48,32)(15,71,49,23)(16,62,50,34)(17,73,51,25)(18,64,52,36)(19,75,53,27)(20,66,54,38) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,43),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,80)], [(1,77,55,29),(2,68,56,40),(3,79,57,31),(4,70,58,22),(5,61,59,33),(6,72,60,24),(7,63,41,35),(8,74,42,26),(9,65,43,37),(10,76,44,28),(11,67,45,39),(12,78,46,30),(13,69,47,21),(14,80,48,32),(15,71,49,23),(16,62,50,34),(17,73,51,25),(18,64,52,36),(19,75,53,27),(20,66,54,38)]])
D20:8C4 is a maximal subgroup of
D20:C8 Dic5:4D8 D4:D5:6C4 D20:3D4 D20.D4 Dic5:7SD16 Q8:D5:6C4 Dic5:SD16 D20.12D4 Dic5:8SD16 D40:15C4 D20:Q8 D20.Q8 D40:12C4 C40:21(C2xC4) D20:2Q8 D20.2Q8 D20:2C8 D10:2M4(2) C20:M4(2) C10.82+ 1+4 C10.2- 1+4 C10.112+ 1+4 C42:7D10 C42.188D10 C42.95D10 C42.97D10 C4xD4xD5 C42:11D10 Dic10:24D4 C42.114D10 C42.122D10 C4xQ8:2D5 C42.126D10 C42.136D10 C20:(C4oD4) D20:19D4 D20:20D4 C10.472+ 1+4 C22:Q8:25D5 C4:C4:26D10 D20:21D4 D20:22D4 Dic10:22D4 C10.532+ 1+4 C10.202- 1+4 C10.242- 1+4 C10.1212+ 1+4 C4:C4:28D10 C10.612+ 1+4 C10.642+ 1+4 D20:7Q8 C42.237D10 C42.150D10 C42.151D10 C42.153D10 C42.156D10 C42:23D10 C42:24D10 C42.189D10 C42.163D10 C42.240D10 D20:8Q8 D20:9Q8 C42.178D10 Dic15:13D4 D60:17C4 D20:8Dic3 C15:20(C4xD4) D60:11C4
D20:8C4 is a maximal quotient of
C4:Dic5:15C4 (C2xC4):9D20 D10:2C42 C10.54(C4xD4) D20:5C8 D10:5M4(2) C20:6M4(2) Dic5:8SD16 Dic20:15C4 D40:15C4 D40:12C4 Dic5:5Q16 C40:21(C2xC4) D40:16C4 D40:13C4 C20:4(C4:C4) C4:C4xDic5 (C2xD20):22C4 D10:5(C4:C4) C10.90(C4xD4) Dic15:13D4 D60:17C4 D20:8Dic3 C15:20(C4xD4) D60:11C4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4oD4 | D10 | C4xD5 | D4xD5 | Q8:2D5 |
| kernel | D20:8C4 | C4xDic5 | D10:C4 | C5xC4:C4 | C2xC4xD5 | C2xD20 | D20 | Dic5 | C4:C4 | C10 | C2xC4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of D20:8C4 ►in GL5(F41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 6 | 40 | 0 | 0 |
| 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 21 |
| 0 | 0 | 0 | 37 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 37 | 1 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 16 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,36,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,21,1],[40,0,0,0,0,0,40,0,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,0,1],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,16,32] >;
D20:8C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_8C_4 % in TeX
G:=Group("D20:8C4"); // GroupNames label
G:=SmallGroup(160,114);
// by ID
G=gap.SmallGroup(160,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,188,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations