metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D20, C42⋊6D5, C20.28D4, (C4×C20)⋊5C2, C2.6(C2×D20), C10.4(C2×D4), (C2×D20).3C2, (C2×C4).77D10, C5⋊1(C4.4D4), D10⋊C4⋊1C2, (C2×Dic10)⋊1C2, C2.7(C4○D20), C10.5(C4○D4), (C2×C20).74C22, (C2×C10).16C23, (C2×Dic5).3C22, (C22×D5).2C22, C22.37(C22×D5), SmallGroup(160,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D20
G = < a,b,c | a4=b20=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 288 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D10⋊C4, C4×C20, C2×Dic10, C2×D20, C4.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, D20, C22×D5, C2×D20, C4○D20, C4.D20
►On 80 points
(1 49 66 38)(2 50 67 39)(3 51 68 40)(4 52 69 21)(5 53 70 22)(6 54 71 23)(7 55 72 24)(8 56 73 25)(9 57 74 26)(10 58 75 27)(11 59 76 28)(12 60 77 29)(13 41 78 30)(14 42 79 31)(15 43 80 32)(16 44 61 33)(17 45 62 34)(18 46 63 35)(19 47 64 36)(20 48 65 37)
(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 65 66 20)(2 19 67 64)(3 63 68 18)(4 17 69 62)(5 61 70 16)(6 15 71 80)(7 79 72 14)(8 13 73 78)(9 77 74 12)(10 11 75 76)(21 45 52 34)(22 33 53 44)(23 43 54 32)(24 31 55 42)(25 41 56 30)(26 29 57 60)(27 59 58 28)(35 51 46 40)(36 39 47 50)(37 49 48 38)
G:=sub<Sym(80)| (1,49,66,38)(2,50,67,39)(3,51,68,40)(4,52,69,21)(5,53,70,22)(6,54,71,23)(7,55,72,24)(8,56,73,25)(9,57,74,26)(10,58,75,27)(11,59,76,28)(12,60,77,29)(13,41,78,30)(14,42,79,31)(15,43,80,32)(16,44,61,33)(17,45,62,34)(18,46,63,35)(19,47,64,36)(20,48,65,37), (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), (1,65,66,20)(2,19,67,64)(3,63,68,18)(4,17,69,62)(5,61,70,16)(6,15,71,80)(7,79,72,14)(8,13,73,78)(9,77,74,12)(10,11,75,76)(21,45,52,34)(22,33,53,44)(23,43,54,32)(24,31,55,42)(25,41,56,30)(26,29,57,60)(27,59,58,28)(35,51,46,40)(36,39,47,50)(37,49,48,38)>;
G:=Group( (1,49,66,38)(2,50,67,39)(3,51,68,40)(4,52,69,21)(5,53,70,22)(6,54,71,23)(7,55,72,24)(8,56,73,25)(9,57,74,26)(10,58,75,27)(11,59,76,28)(12,60,77,29)(13,41,78,30)(14,42,79,31)(15,43,80,32)(16,44,61,33)(17,45,62,34)(18,46,63,35)(19,47,64,36)(20,48,65,37), (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), (1,65,66,20)(2,19,67,64)(3,63,68,18)(4,17,69,62)(5,61,70,16)(6,15,71,80)(7,79,72,14)(8,13,73,78)(9,77,74,12)(10,11,75,76)(21,45,52,34)(22,33,53,44)(23,43,54,32)(24,31,55,42)(25,41,56,30)(26,29,57,60)(27,59,58,28)(35,51,46,40)(36,39,47,50)(37,49,48,38) );
G=PermutationGroup([[(1,49,66,38),(2,50,67,39),(3,51,68,40),(4,52,69,21),(5,53,70,22),(6,54,71,23),(7,55,72,24),(8,56,73,25),(9,57,74,26),(10,58,75,27),(11,59,76,28),(12,60,77,29),(13,41,78,30),(14,42,79,31),(15,43,80,32),(16,44,61,33),(17,45,62,34),(18,46,63,35),(19,47,64,36),(20,48,65,37)], [(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)], [(1,65,66,20),(2,19,67,64),(3,63,68,18),(4,17,69,62),(5,61,70,16),(6,15,71,80),(7,79,72,14),(8,13,73,78),(9,77,74,12),(10,11,75,76),(21,45,52,34),(22,33,53,44),(23,43,54,32),(24,31,55,42),(25,41,56,30),(26,29,57,60),(27,59,58,28),(35,51,46,40),(36,39,47,50),(37,49,48,38)]])
C4.D20 is a maximal subgroup of
C42.D10 C42⋊2F5 C42.2F5 C8.8D20 C42.264D10 C8⋊D20 C42.20D10 C8.D20 M4(2)⋊D10 D4.10D20 D20.19D4 C42.36D10 D4.1D20 Q8.1D20 C42.214D10 C42.216D10 C42.74D10 C42.80D10 C42.82D10 C42.276D10 C42.277D10 C42⋊9D10 C42.92D10 C42⋊10D10 C42.97D10 C42.99D10 D20⋊23D4 Dic10⋊23D4 D4⋊5D20 C42.114D10 C42⋊17D10 C42.122D10 Q8⋊5D20 C42.133D10 C42.135D10 C42.136D10 C42.233D10 D5×C4.4D4 C42⋊22D10 C42.145D10 C42.237D10 C42.157D10 C42.158D10 C42⋊23D10 C42.164D10 C42⋊26D10 C42.171D10 C42.178D10 Dic3.D20 C60.44D4 C60.47D4 C42⋊7D15
C4.D20 is a maximal quotient of
(C2×C20).28D4 (C2×Dic5)⋊3D4 (C2×C4).20D20 (C2×C4).21D20 C20.14Q16 C4.5D40 C42.264D10 C42.14D10 C42.19D10 C42.20D10 (C2×C20)⋊10Q8 C42⋊9Dic5 (C2×C4)⋊6D20 (C2×C42)⋊D5 Dic3.D20 C60.44D4 C60.47D4 C42⋊7D15
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D20 | C4○D20 |
| kernel | C4.D20 | D10⋊C4 | C4×C20 | C2×Dic10 | C2×D20 | C20 | C42 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 8 | 16 |
Matrix representation of C4.D20 ►in GL4(𝔽41) generated by
| 2 | 13 | 0 | 0 |
| 28 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 23 | 0 | 0 |
| 18 | 21 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 6 | 23 | 0 | 0 |
| 18 | 35 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [2,28,0,0,13,39,0,0,0,0,40,0,0,0,0,40],[6,18,0,0,23,21,0,0,0,0,0,1,0,0,40,0],[6,18,0,0,23,35,0,0,0,0,0,1,0,0,1,0] >;
C4.D20 in GAP, Magma, Sage, TeX
C_4.D_{20} % in TeX
G:=Group("C4.D20"); // GroupNames label
G:=SmallGroup(160,96);
// by ID
G=gap.SmallGroup(160,96);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,55,218,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations