direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4xC10, C23:C10, C20:4C22, C10.11C23, C4:(C2xC10), (C2xC20):6C2, (C2xC4):2C10, C22:(C2xC10), (C2xC10):2C22, (C22xC10):1C2, C2.1(C22xC10), SmallGroup(80,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4xC10
G = < a,b,c | a10=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2xC4, D4, C23, C10, C10, C10, C2xD4, C20, C2xC10, C2xC10, C2xC10, C2xC20, C5xD4, C22xC10, D4xC10
Quotients: C1, C2, C22, C5, D4, C23, C10, C2xD4, C2xC10, C5xD4, C22xC10, D4xC10
►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 38 26 19)(2 39 27 20)(3 40 28 11)(4 31 29 12)(5 32 30 13)(6 33 21 14)(7 34 22 15)(8 35 23 16)(9 36 24 17)(10 37 25 18)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 21)(20 22)
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), (1,38,26,19)(2,39,27,20)(3,40,28,11)(4,31,29,12)(5,32,30,13)(6,33,21,14)(7,34,22,15)(8,35,23,16)(9,36,24,17)(10,37,25,18), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22)>;
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), (1,38,26,19)(2,39,27,20)(3,40,28,11)(4,31,29,12)(5,32,30,13)(6,33,21,14)(7,34,22,15)(8,35,23,16)(9,36,24,17)(10,37,25,18), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22) );
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)], [(1,38,26,19),(2,39,27,20),(3,40,28,11),(4,31,29,12),(5,32,30,13),(6,33,21,14),(7,34,22,15),(8,35,23,16),(9,36,24,17),(10,37,25,18)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,21),(20,22)]])
D4xC10 is a maximal subgroup of
D4:Dic5 C20.D4 C23:Dic5 D4.D10 C23.18D10 C20.17D4 C23:D10 C20:2D4 Dic5:D4 C20:D4 D4:6D10
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | C5xD4 |
| kernel | D4xC10 | C2xC20 | C5xD4 | C22xC10 | C2xD4 | C2xC4 | D4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 2 | 8 |
Matrix representation of D4xC10 ►in GL3(F41) generated by
| 40 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 40 | 0 |
G:=sub<GL(3,GF(41))| [40,0,0,0,18,0,0,0,18],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;
D4xC10 in GAP, Magma, Sage, TeX
D_4\times C_{10} % in TeX
G:=Group("D4xC10"); // GroupNames label
G:=SmallGroup(80,46);
// by ID
G=gap.SmallGroup(80,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-2,421]);
// Polycyclic
G:=Group<a,b,c|a^10=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations