direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D5, C20⋊3C22, C10.2C23, C22.9D10, Dic5⋊3C22, D10.8C22, C4○(C4×D5), C10⋊2(C2×C4), (C2×C20)⋊5C2, C5⋊2(C22×C4), C4○(C2×Dic5), (C2×C4)○Dic5, (C2×Dic5)⋊5C2, C2.1(C22×D5), (C2×C10).9C22, (C22×D5).4C2, (C2×C4)○(C2×Dic5), SmallGroup(80,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C4×D5 |
Generators and relations for C2×C4×D5
G = < a,b,c,d | a2=b4=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 130 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C10, C22×C4, Dic5, C20, D10, C2×C10, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C4×D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, C2×C4×D5
►On 40 points
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)
G:=sub<Sym(40)| (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)>;
G:=Group( (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39) );
G=PermutationGroup([[(1,24),(2,25),(3,21),(4,22),(5,23),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39)]])
C2×C4×D5 is a maximal subgroup of
D10⋊1C8 D10⋊C8 D10.3Q8 C42⋊D5 Dic5⋊4D4 D10.12D4 D10⋊D4 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊Q8 D10⋊2Q8 C20⋊2D4 D10⋊3Q8 D5⋊M4(2) D10.C23
C2×C4×D5 is a maximal quotient of
C42⋊D5 C23.11D10 Dic5⋊4D4 Dic5⋊3Q8 C4⋊C4⋊7D5 D20⋊8C4 D20.3C4 D20.2C4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | C4×D5 |
| kernel | C2×C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 8 |
Matrix representation of C2×C4×D5 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 40 | 1 |
| 0 | 5 | 35 |
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 5 | 1 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,32,0,0,0,32],[1,0,0,0,40,5,0,1,35],[40,0,0,0,40,5,0,0,1] >;
C2×C4×D5 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_5
% in TeX
G:=Group("C2xC4xD5"); // GroupNames label
G:=SmallGroup(80,36);
// by ID
G=gap.SmallGroup(80,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,42,1604]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations