direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×F5, C10⋊C42, D5⋊C42, D10.9C23, C5⋊(C2×C42), C20⋊3(C2×C4), (C2×C20)⋊5C4, (C4×D5)⋊5C4, D5.(C22×C4), (C2×Dic5)⋊7C4, Dic5⋊6(C2×C4), C2.2(C22×F5), D10.15(C2×C4), C10.4(C22×C4), (C2×F5).5C22, (C22×F5).3C2, C22.17(C2×F5), (C4×D5).35C22, (C22×D5).36C22, (C2×C4×D5).18C2, (C2×C10).16(C2×C4), SmallGroup(160,203)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C4×F5 |
Generators and relations for C2×C4×F5
G = < a,b,c,d | a2=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 292 in 108 conjugacy classes, 62 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C10, C42, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×F5, C2×C4×D5, C22×F5, C2×C4×F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, F5, C2×C42, C2×F5, C4×F5, C22×F5, C2×C4×F5
►On 40 points
(1 21)(2 22)(3 23)(4 24)(5 25)(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 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(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 31 6 36)(2 33 10 39)(3 35 9 37)(4 32 8 40)(5 34 7 38)(11 26 16 21)(12 28 20 24)(13 30 19 22)(14 27 18 25)(15 29 17 23)
G:=sub<Sym(40)| (1,21)(2,22)(3,23)(4,24)(5,25)(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), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(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), (1,31,6,36)(2,33,10,39)(3,35,9,37)(4,32,8,40)(5,34,7,38)(11,26,16,21)(12,28,20,24)(13,30,19,22)(14,27,18,25)(15,29,17,23)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(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), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(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), (1,31,6,36)(2,33,10,39)(3,35,9,37)(4,32,8,40)(5,34,7,38)(11,26,16,21)(12,28,20,24)(13,30,19,22)(14,27,18,25)(15,29,17,23) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(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)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(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)], [(1,31,6,36),(2,33,10,39),(3,35,9,37),(4,32,8,40),(5,34,7,38),(11,26,16,21),(12,28,20,24),(13,30,19,22),(14,27,18,25),(15,29,17,23)]])
C2×C4×F5 is a maximal subgroup of
C20.C42 D10.3M4(2) M4(2)⋊3F5 C42⋊4F5 C10.(C4×D4) C4⋊C4⋊5F5 C20⋊(C4⋊C4) C2.(D4×F5) (C2×F5)⋊Q8 D5⋊C4≀C2
C2×C4×F5 is a maximal quotient of
C42.5F5 C42⋊4F5 Dic5.C42 D10.C42 C20.12C42 M4(2)⋊5F5
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4X | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | F5 | C2×F5 | C2×F5 | C4×F5 |
| kernel | C2×C4×F5 | C4×F5 | C2×C4×D5 | C22×F5 | C4×D5 | C2×Dic5 | C2×C20 | C2×F5 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 16 | 1 | 2 | 1 | 4 |
Matrix representation of C2×C4×F5 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 1 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,9,0,0,0,0,0,0,9,0] >;
C2×C4×F5 in GAP, Magma, Sage, TeX
C_2\times C_4\times F_5
% in TeX
G:=Group("C2xC4xF5"); // GroupNames label
G:=SmallGroup(160,203);
// by ID
G=gap.SmallGroup(160,203);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,2309,599]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations