metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20:23D4, C42:14D10, C10.1032+ 1+4, C4:C4:47D10, (C4xD4):12D5, (D4xC20):14C2, (C4xD20):28C2, C5:2(D4:5D4), C4.139(D4xD5), D10:D4:7C2, C20:7D4:18C2, (C4xC20):19C22, C22:C4:46D10, D10.37(C2xD4), C20.345(C2xD4), (C22xD20):9C2, (C22xC4):12D10, C23:D10:20C2, (C2xD4).213D10, C4.D20:16C2, C22:3(C4oD20), (C2xC10).94C24, C4:Dic5:59C22, C10.49(C22xD4), D10.13D4:7C2, C20.48D4:10C2, (C2xC20).782C23, (C22xC20):16C22, C10.D4:3C22, C2.15(D4:8D10), D10:C4:30C22, C23.94(C22xD5), (C2xDic10):53C22, (C2xD20).218C22, (D4xC10).305C22, (C2xDic5).40C23, (C23xD5).39C22, C22.119(C23xD5), C23.D5.11C22, (C22xC10).164C23, (C22xD5).182C23, C2.22(C2xD4xD5), (C2xC4oD20):7C2, (C2xC4xD5):48C22, (C2xC10):2(C4oD4), (C5xC4:C4):59C22, (D5xC22:C4):28C2, C2.45(C2xC4oD20), C10.41(C2xC4oD4), (C2xC5:D4):3C22, (C5xC22:C4):56C22, (C2xC4).158(C22xD5), SmallGroup(320,1222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20:23D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd=c-1 >
Subgroups: 1462 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C2xC22:C4, C4xD4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xD4, C2xC4oD4, Dic10, C4xD5, D20, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, D4:5D4, C10.D4, C4:Dic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C2xDic10, C2xC4xD5, C2xD20, C2xD20, C2xD20, C4oD20, C2xC5:D4, C22xC20, D4xC10, C23xD5, C4xD20, C4.D20, D5xC22:C4, D10:D4, D10.13D4, C20.48D4, C20:7D4, C23:D10, D4xC20, C22xD20, C2xC4oD20, D20:23D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C22xD4, C2xC4oD4, 2+ 1+4, C22xD5, D4:5D4, C4oD20, D4xD5, C23xD5, C2xC4oD20, C2xD4xD5, D4:8D10, D20:23D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 21 79 43)(2 22 80 44)(3 23 61 45)(4 24 62 46)(5 25 63 47)(6 26 64 48)(7 27 65 49)(8 28 66 50)(9 29 67 51)(10 30 68 52)(11 31 69 53)(12 32 70 54)(13 33 71 55)(14 34 72 56)(15 35 73 57)(16 36 74 58)(17 37 75 59)(18 38 76 60)(19 39 77 41)(20 40 78 42)
(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,21,79,43)(2,22,80,44)(3,23,61,45)(4,24,62,46)(5,25,63,47)(6,26,64,48)(7,27,65,49)(8,28,66,50)(9,29,67,51)(10,30,68,52)(11,31,69,53)(12,32,70,54)(13,33,71,55)(14,34,72,56)(15,35,73,57)(16,36,74,58)(17,37,75,59)(18,38,76,60)(19,39,77,41)(20,40,78,42), (21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,21,79,43)(2,22,80,44)(3,23,61,45)(4,24,62,46)(5,25,63,47)(6,26,64,48)(7,27,65,49)(8,28,66,50)(9,29,67,51)(10,30,68,52)(11,31,69,53)(12,32,70,54)(13,33,71,55)(14,34,72,56)(15,35,73,57)(16,36,74,58)(17,37,75,59)(18,38,76,60)(19,39,77,41)(20,40,78,42), (21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79)], [(1,21,79,43),(2,22,80,44),(3,23,61,45),(4,24,62,46),(5,25,63,47),(6,26,64,48),(7,27,65,49),(8,28,66,50),(9,29,67,51),(10,30,68,52),(11,31,69,53),(12,32,70,54),(13,33,71,55),(14,34,72,56),(15,35,73,57),(16,36,74,58),(17,37,75,59),(18,38,76,60),(19,39,77,41),(20,40,78,42)], [(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,41),(40,42)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4oD4 | D10 | D10 | D10 | D10 | D10 | C4oD20 | 2+ 1+4 | D4xD5 | D4:8D10 |
| kernel | D20:23D4 | C4xD20 | C4.D20 | D5xC22:C4 | D10:D4 | D10.13D4 | C20.48D4 | C20:7D4 | C23:D10 | D4xC20 | C22xD20 | C2xC4oD20 | D20 | C4xD4 | C2xC10 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | C22 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D20:23D4 ►in GL4(F41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 11 |
| 0 | 0 | 30 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 27 |
| 0 | 0 | 32 | 11 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,14,30,0,0,11,9],[1,0,0,0,0,1,0,0,0,0,30,32,0,0,27,11],[0,1,0,0,40,0,0,0,0,0,17,40,0,0,1,24],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;
D20:23D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{23}D_4 % in TeX
G:=Group("D20:23D4"); // GroupNames label
G:=SmallGroup(320,1222);
// by ID
G=gap.SmallGroup(320,1222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations