metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D20, C23.1D10, C5:3(C23:C4), C22:C4:1D5, (C2xDic5):1C4, (C2xC10).27D4, (C22xD5):1C4, C23.D5:1C2, C22.3(C4xD5), C22.8(C5:D4), C2.4(D10:C4), C10.13(C22:C4), (C22xC10).5C22, (C5xC22:C4):1C2, (C2xC5:D4).1C2, (C2xC10).21(C2xC4), SmallGroup(160,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D20
G = < a,b,c,d | a2=b2=c20=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Quotients: C1, C2 [x3], C4 [x2], C22, C2xC4, D4 [x2], D5, C22:C4, D10, C23:C4, C4xD5, D20, C5:D4, D10:C4, C22.2D20
2C2
2C2
2C2
20C2
4C4
4C22
10C22
10C4
20C4
20C22
2C10
2C10
2C10
4D5
5C23
10C2xC4
10D4
10D4
2Dic5
2D10
4Dic5
4D10
4C20
Smallest permutation representation of C22.2D20
►On 40 points
Generators in S40
(1 37)(3 39)(5 21)(7 23)(9 25)(11 27)(13 29)(15 31)(17 33)(19 35)
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)
(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)
(1 31 37 15)(2 30)(3 13 39 29)(4 12)(5 27 21 11)(6 26)(7 9 23 25)(10 22)(14 38)(16 20)(17 35 33 19)(18 34)(28 40)(32 36)
G:=sub<Sym(40)| (1,37)(3,39)(5,21)(7,23)(9,25)(11,27)(13,29)(15,31)(17,33)(19,35), (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36), (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), (1,31,37,15)(2,30)(3,13,39,29)(4,12)(5,27,21,11)(6,26)(7,9,23,25)(10,22)(14,38)(16,20)(17,35,33,19)(18,34)(28,40)(32,36)>;
G:=Group( (1,37)(3,39)(5,21)(7,23)(9,25)(11,27)(13,29)(15,31)(17,33)(19,35), (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36), (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), (1,31,37,15)(2,30)(3,13,39,29)(4,12)(5,27,21,11)(6,26)(7,9,23,25)(10,22)(14,38)(16,20)(17,35,33,19)(18,34)(28,40)(32,36) );
G=PermutationGroup([(1,37),(3,39),(5,21),(7,23),(9,25),(11,27),(13,29),(15,31),(17,33),(19,35)], [(1,37),(2,38),(3,39),(4,40),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36)], [(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)], [(1,31,37,15),(2,30),(3,13,39,29),(4,12),(5,27,21,11),(6,26),(7,9,23,25),(10,22),(14,38),(16,20),(17,35,33,19),(18,34),(28,40),(32,36)])
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | C4xD5 | D20 | C5:D4 | C23:C4 | C22.2D20 |
| kernel | C22.2D20 | C23.D5 | C5xC22:C4 | C2xC5:D4 | C2xDic5 | C22xD5 | C2xC10 | C22:C4 | C23 | C22 | C22 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 4 |
Matrix representation of C22.2D20 ►in GL4(F41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 6 |
| 35 | 18 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 23 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,35,23,0,0,18,20,0,40,0,0,1,6,0,0],[35,23,0,0,18,6,0,0,0,0,0,1,0,0,1,0] >;
C22.2D20 in GAP, Magma, Sage, TeX
C_2^2._2D_{20} % in TeX
G:=Group("C2^2.2D20"); // GroupNames label
G:=SmallGroup(160,13);
// by ID
G=gap.SmallGroup(160,13);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,297,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations