metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20:1D4, C4:2D20, D10:2D4, C4:C4:3D5, (C2xD20):4C2, C5:2(C4:D4), C2.13(D4xD5), C2.9(C2xD20), C10.7(C2xD4), (C2xC4).12D10, D10:C4:8C2, (C2xC20).5C22, C10.34(C4oD4), (C2xC10).36C23, C2.6(Q8:2D5), (C22xD5).7C22, C22.50(C22xD5), (C2xDic5).34C22, (C2xC4xD5):1C2, (C5xC4:C4):6C2, SmallGroup(160,116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4:D20
G = < a,b,c | a4=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 384 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, C23, D5, C10, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, C20, C20, D10, D10, C2xC10, C4:D4, C4xD5, D20, C2xDic5, C2xC20, C2xC20, C22xD5, C22xD5, D10:C4, C5xC4:C4, C2xC4xD5, C2xD20, C2xD20, C4:D20
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, D10, C4:D4, D20, C22xD5, C2xD20, D4xD5, Q8:2D5, C4:D20
►On 80 points
(1 71 41 26)(2 27 42 72)(3 73 43 28)(4 29 44 74)(5 75 45 30)(6 31 46 76)(7 77 47 32)(8 33 48 78)(9 79 49 34)(10 35 50 80)(11 61 51 36)(12 37 52 62)(13 63 53 38)(14 39 54 64)(15 65 55 40)(16 21 56 66)(17 67 57 22)(18 23 58 68)(19 69 59 24)(20 25 60 70)
(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)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)
G:=sub<Sym(80)| (1,71,41,26)(2,27,42,72)(3,73,43,28)(4,29,44,74)(5,75,45,30)(6,31,46,76)(7,77,47,32)(8,33,48,78)(9,79,49,34)(10,35,50,80)(11,61,51,36)(12,37,52,62)(13,63,53,38)(14,39,54,64)(15,65,55,40)(16,21,56,66)(17,67,57,22)(18,23,58,68)(19,69,59,24)(20,25,60,70), (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)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)>;
G:=Group( (1,71,41,26)(2,27,42,72)(3,73,43,28)(4,29,44,74)(5,75,45,30)(6,31,46,76)(7,77,47,32)(8,33,48,78)(9,79,49,34)(10,35,50,80)(11,61,51,36)(12,37,52,62)(13,63,53,38)(14,39,54,64)(15,65,55,40)(16,21,56,66)(17,67,57,22)(18,23,58,68)(19,69,59,24)(20,25,60,70), (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)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59) );
G=PermutationGroup([[(1,71,41,26),(2,27,42,72),(3,73,43,28),(4,29,44,74),(5,75,45,30),(6,31,46,76),(7,77,47,32),(8,33,48,78),(9,79,49,34),(10,35,50,80),(11,61,51,36),(12,37,52,62),(13,63,53,38),(14,39,54,64),(15,65,55,40),(16,21,56,66),(17,67,57,22),(18,23,58,68),(19,69,59,24),(20,25,60,70)], [(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),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59)]])
C4:D20 is a maximal subgroup of
D10.1D8 D4:D20 D10:D8 D4:3D20 C5:(C8:2D4) Q8:2D20 D10:2SD16 D20:4D4 C5:(C8:D4) D10.17SD16 C8:8D20 C8:2D20 C4.Q8:D5 D10.13D8 C8:7D20 C2.D8:D5 C8:3D20 C10.2- 1+4 C10.2+ 1+4 C10.112+ 1+4 C42:8D10 C42:9D10 C42.95D10 C42.97D10 C42.228D10 D4xD20 D4:5D20 C42.116D10 Q8:5D20 Q8:6D20 C42.131D10 C42.133D10 Dic10:20D4 D5xC4:D4 C10.382+ 1+4 D20:19D4 C4:C4:26D10 C10.172- 1+4 D20:21D4 Dic10:22D4 C10.562+ 1+4 C10.262- 1+4 C10.1202+ 1+4 C10.1212+ 1+4 C10.662+ 1+4 C10.682+ 1+4 C42.237D10 C42.150D10 C42.153D10 C42.156D10 C42.158D10 C42:23D10 C42.163D10 C42:25D10 C42.240D10 D20:12D4 C42.178D10 C42.179D10 Dic3:D20 C12:7D20 D30:2D4 C12:2D20 C4:D60
C4:D20 is a maximal quotient of
C10.(C4:Q8) (C2xC4):9D20 C10.55(C4xD4) (C2xC20):5D4 (C2xDic5):3D4 (C2xC4).21D20 C20:SD16 C4:D40 D20.19D4 C42.36D10 Dic10:8D4 C4:Dic20 C8:8D20 C8:2D20 C8.2D20 C8:7D20 C8:3D20 D10:2Q16 C8.20D20 C8.21D20 C8.24D20 C20:6(C4:C4) D10:4(C4:C4) (C2xD20):22C4 (C2xC4):3D20 (C2xC20).56D4 Dic3:D20 C12:7D20 D30:2D4 C12:2D20 C4:D60
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4oD4 | D10 | D20 | D4xD5 | Q8:2D5 |
| kernel | C4:D20 | D10:C4 | C5xC4:C4 | C2xC4xD5 | C2xD20 | C20 | D10 | C4:C4 | C10 | C2xC4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of C4:D20 ►in GL4(F41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 14 | 39 | 0 | 0 |
| 16 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[14,16,0,0,39,30,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,1,1,0,0,0,40] >;
C4:D20 in GAP, Magma, Sage, TeX
C_4\rtimes D_{20} % in TeX
G:=Group("C4:D20"); // GroupNames label
G:=SmallGroup(160,116);
// by ID
G=gap.SmallGroup(160,116);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations