direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)×F5, C20.6C42, D10.2C42, Dic5.2C42, C8⋊6(C2×F5), C40⋊6(C2×C4), (C8×F5)⋊7C2, C4.F5⋊3C4, C8⋊D5⋊2C4, C8⋊F5⋊6C2, C5⋊2(C4×M4(2)), (C4×F5).3C4, C4.11(C4×F5), C4.Dic5⋊4C4, C22.F5⋊6C4, (C5×M4(2))⋊5C4, (C2×C10).6C42, (C22×F5).4C4, C22.11(C4×F5), C4.52(C22×F5), C20.92(C22×C4), C10.14(C2×C42), D5⋊C8.19C22, D5⋊M4(2).3C2, (C4×D5).88C23, (C8×D5).36C22, D5.2(C2×M4(2)), (C4×F5).18C22, D10.34(C22×C4), (D5×M4(2)).10C2, Dic5.33(C22×C4), C5⋊C8⋊2(C2×C4), (C2×C4×F5).4C2, C2.15(C2×C4×F5), C5⋊2C8⋊16(C2×C4), (C2×F5).7(C2×C4), (C2×C4).76(C2×F5), (C2×C20).45(C2×C4), (C4×D5).46(C2×C4), (C2×C4×D5).192C22, (C2×Dic5).66(C2×C4), (C22×D5).52(C2×C4), SmallGroup(320,1064)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 442 in 142 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×13], C23, D5 [×2], D5, C10, C10, C42 [×4], C2×C8 [×4], M4(2), M4(2) [×7], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], F5 [×2], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C5⋊2C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C4×M4(2), C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), D5⋊C8 [×2], C4.F5 [×2], C4×F5 [×2], C4×F5 [×2], C22.F5 [×2], C2×C4×D5, C22×F5 [×2], C8×F5 [×2], C8⋊F5 [×2], D5×M4(2), D5⋊M4(2), C2×C4×F5, M4(2)×F5
Quotients:
C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C2×C42, C2×M4(2) [×2], C2×F5 [×3], C4×M4(2), C4×F5 [×2], C22×F5, C2×C4×F5, M4(2)×F5
Generators and relations
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a5, 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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)
(1 10 19 37 26)(2 11 20 38 27)(3 12 21 39 28)(4 13 22 40 29)(5 14 23 33 30)(6 15 24 34 31)(7 16 17 35 32)(8 9 18 36 25)
(1 7 5 3)(2 8 6 4)(9 24 29 38)(10 17 30 39)(11 18 31 40)(12 19 32 33)(13 20 25 34)(14 21 26 35)(15 22 27 36)(16 23 28 37)
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40), (1,10,19,37,26)(2,11,20,38,27)(3,12,21,39,28)(4,13,22,40,29)(5,14,23,33,30)(6,15,24,34,31)(7,16,17,35,32)(8,9,18,36,25), (1,7,5,3)(2,8,6,4)(9,24,29,38)(10,17,30,39)(11,18,31,40)(12,19,32,33)(13,20,25,34)(14,21,26,35)(15,22,27,36)(16,23,28,37)>;
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40), (1,10,19,37,26)(2,11,20,38,27)(3,12,21,39,28)(4,13,22,40,29)(5,14,23,33,30)(6,15,24,34,31)(7,16,17,35,32)(8,9,18,36,25), (1,7,5,3)(2,8,6,4)(9,24,29,38)(10,17,30,39)(11,18,31,40)(12,19,32,33)(13,20,25,34)(14,21,26,35)(15,22,27,36)(16,23,28,37) );
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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40)], [(1,10,19,37,26),(2,11,20,38,27),(3,12,21,39,28),(4,13,22,40,29),(5,14,23,33,30),(6,15,24,34,31),(7,16,17,35,32),(8,9,18,36,25)], [(1,7,5,3),(2,8,6,4),(9,24,29,38),(10,17,30,39),(11,18,31,40),(12,19,32,33),(13,20,25,34),(14,21,26,35),(15,22,27,36),(16,23,28,37)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 39 | 0 | 0 | 0 | 0 |
| 37 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 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 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
G:=sub<GL(6,GF(41))| [40,37,0,0,0,0,39,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | ··· | 4M | 4N | ··· | 4R | 5 | 8A | 8B | 8C | 8D | 8E | ··· | 8P | 10A | 10B | 20A | 20B | 20C | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4 | M4(2) | F5 | C2×F5 | C2×F5 | C4×F5 | C4×F5 | M4(2)×F5 |
| kernel | M4(2)×F5 | C8×F5 | C8⋊F5 | D5×M4(2) | D5⋊M4(2) | C2×C4×F5 | C8⋊D5 | C4.Dic5 | C5×M4(2) | C4.F5 | C4×F5 | C22.F5 | C22×F5 | F5 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
M_{4(2)}\times F_5 % in TeX
G:=Group("M4(2)xF5"); // GroupNames label
G:=SmallGroup(320,1064);
// by ID
G=gap.SmallGroup(320,1064);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations