direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C10, C23⋊C10, C20⋊4C22, C10.11C23, C4⋊(C2×C10), (C2×C20)⋊6C2, (C2×C4)⋊2C10, C22⋊(C2×C10), (C2×C10)⋊2C22, (C22×C10)⋊1C2, C2.1(C22×C10), SmallGroup(80,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C10
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, C2×C4, D4, C23, C10, C10, C10, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×C20, C5×D4, C22×C10, D4×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C5×D4, C22×C10, D4×C10
►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)]])
D4×C10 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 | C5×D4 |
| kernel | D4×C10 | C2×C20 | C5×D4 | C22×C10 | C2×D4 | C2×C4 | D4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 2 | 8 |
Matrix representation of D4×C10 ►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] >;
D4×C10 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