direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8×F5, D40⋊5C4, C5⋊(C4×D8), C8⋊4(C2×F5), C40⋊2(C2×C4), D4⋊D5⋊1C4, (C5×D8)⋊5C4, (D4×F5)⋊1C2, (C8×F5)⋊1C2, D4⋊1(C2×F5), D20⋊1(C2×C4), D5.D8⋊2C2, D5.2(C2×D8), (D5×D8).5C2, C2.15(D4×F5), D20⋊C4⋊1C2, C10.14(C4×D4), (C2×F5).10D4, C4⋊F5.1C22, C4.1(C22×F5), D5.2(C4○D8), D10.63(C2×D4), C20.1(C22×C4), D5⋊C8.9C22, (D4×D5).5C22, (C4×F5).9C22, (C4×D5).23C23, (C8×D5).21C22, Dic5.1(C4○D4), (C5×D4)⋊1(C2×C4), C5⋊2C8⋊11(C2×C4), SmallGroup(320,1068)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 674 in 134 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×6], C4, C4 [×6], C22 [×9], C5, C8, C8 [×2], C2×C4 [×9], D4 [×2], D4 [×4], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8, D8 [×3], C22×C4 [×2], C2×D4 [×2], Dic5, C20, F5 [×2], F5 [×3], D10, D10 [×6], C2×C10 [×2], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C5⋊2C8, C40, C5⋊C8, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5 [×2], C2×F5 [×6], C22×D5 [×2], C4×D8, C8×D5, D40, D4⋊D5 [×2], C5×D8, D5⋊C8, C4×F5, C4⋊F5 [×2], C22⋊F5 [×2], D4×D5 [×2], C22×F5 [×2], C8×F5, D5.D8, D20⋊C4 [×2], D5×D8, D4×F5 [×2], D8×F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×D8, C4○D8, C2×F5 [×3], C4×D8, C22×F5, D4×F5, D8×F5
Generators and relations
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)
(1 26 19 12 35)(2 27 20 13 36)(3 28 21 14 37)(4 29 22 15 38)(5 30 23 16 39)(6 31 24 9 40)(7 32 17 10 33)(8 25 18 11 34)
(9 31 24 40)(10 32 17 33)(11 25 18 34)(12 26 19 35)(13 27 20 36)(14 28 21 37)(15 29 22 38)(16 30 23 39)
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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,39), (1,26,19,12,35)(2,27,20,13,36)(3,28,21,14,37)(4,29,22,15,38)(5,30,23,16,39)(6,31,24,9,40)(7,32,17,10,33)(8,25,18,11,34), (9,31,24,40)(10,32,17,33)(11,25,18,34)(12,26,19,35)(13,27,20,36)(14,28,21,37)(15,29,22,38)(16,30,23,39)>;
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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,39), (1,26,19,12,35)(2,27,20,13,36)(3,28,21,14,37)(4,29,22,15,38)(5,30,23,16,39)(6,31,24,9,40)(7,32,17,10,33)(8,25,18,11,34), (9,31,24,40)(10,32,17,33)(11,25,18,34)(12,26,19,35)(13,27,20,36)(14,28,21,37)(15,29,22,38)(16,30,23,39) );
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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,20),(18,19),(21,24),(22,23),(25,26),(27,32),(28,31),(29,30),(33,36),(34,35),(37,40),(38,39)], [(1,26,19,12,35),(2,27,20,13,36),(3,28,21,14,37),(4,29,22,15,38),(5,30,23,16,39),(6,31,24,9,40),(7,32,17,10,33),(8,25,18,11,34)], [(9,31,24,40),(10,32,17,33),(11,25,18,34),(12,26,19,35),(13,27,20,36),(14,28,21,37),(15,29,22,38),(16,30,23,39)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 24 | 0 | 0 | 0 | 0 |
| 29 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 24 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,29,0,0,0,0,24,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,24,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | 8B | 8C | ··· | 8H | 10A | 10B | 10C | 20 | 40A | 40B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 40 | 40 |
| size | 1 | 1 | 4 | 4 | 5 | 5 | 20 | 20 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 4 | 2 | 2 | 10 | ··· | 10 | 4 | 16 | 16 | 8 | 8 | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | D8 | C4○D8 | F5 | C2×F5 | C2×F5 | D4×F5 | D8×F5 |
| kernel | D8×F5 | C8×F5 | D5.D8 | D20⋊C4 | D5×D8 | D4×F5 | D40 | D4⋊D5 | C5×D8 | C2×F5 | Dic5 | F5 | D5 | D8 | C8 | D4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_8\times F_5
% in TeX
G:=Group("D8xF5"); // GroupNames label
G:=SmallGroup(320,1068);
// by ID
G=gap.SmallGroup(320,1068);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations