metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20:7C4, C20.54D4, Dic10:7C4, M4(2):4D5, C22.3D20, C5:4C4wrC2, C4.3(C4xD5), (C2xC10).1D4, C20.27(C2xC4), (C4xDic5):1C2, C4oD20.2C2, (C2xC4).38D10, C4.29(C5:D4), (C5xM4(2)):8C2, (C2xC20).15C22, C10.21(C22:C4), C2.11(D10:C4), SmallGroup(160,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20:7C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a3b >
Quotients: C1, C2, C4, C22, C2xC4, D4, D5, C22:C4, D10, C4wrC2, C4xD5, D20, C5:D4, D10:C4, D20:7C4
Smallest permutation representation of D20:7C4
►On 40 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)
(1 22)(2 21)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 30 31 40)(22 39 32 29)(23 28 33 38)(24 37 34 27)(25 26 35 36)
G:=sub<Sym(40)| (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), (1,22)(2,21)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,30,31,40)(22,39,32,29)(23,28,33,38)(24,37,34,27)(25,26,35,36)>;
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), (1,22)(2,21)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,30,31,40)(22,39,32,29)(23,28,33,38)(24,37,34,27)(25,26,35,36) );
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)], [(1,22),(2,21),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,30,31,40),(22,39,32,29),(23,28,33,38),(24,37,34,27),(25,26,35,36)]])
D20:7C4 is a maximal subgroup of
D20.1D4 D20:1D4 D20.4D4 D20.5D4 D5xC4wrC2 C42:D10 D40:16C4 D40:13C4 C23.20D20 C40.93D4 C40.50D4 D20:18D4 D20.38D4 D20.39D4 D20.40D4 C60.96D4 D60:16C4 D60:10C4
D20:7C4 is a maximal quotient of
C42.D10 C42.2D10 C23.30D20 C22.2D40 D20:4C8 Dic10:4C8 C20.33C42 C60.96D4 D60:16C4 D60:10C4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D10 | C4wrC2 | C4xD5 | C5:D4 | D20 | D20:7C4 |
| kernel | D20:7C4 | C4xDic5 | C5xM4(2) | C4oD20 | Dic10 | D20 | C20 | C2xC10 | M4(2) | C2xC4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
Matrix representation of D20:7C4 ►in GL4(F41) generated by
| 7 | 1 | 0 | 0 |
| 33 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 0 |
| 34 | 35 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [7,33,0,0,1,40,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,1,40,0,0,0,0,0,9,0,0,32,0],[34,8,0,0,35,7,0,0,0,0,40,0,0,0,0,9] >;
D20:7C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_7C_4 % in TeX
G:=Group("D20:7C4"); // GroupNames label
G:=SmallGroup(160,32);
// by ID
G=gap.SmallGroup(160,32);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,86,579,297,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^3*b>;
// generators/relations
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