direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×M4(2), C8⋊6D10, C40⋊6C22, C20.38C23, (C8×D5)⋊7C2, C8⋊D5⋊5C2, (C4×D5).1C4, C4.15(C4×D5), C5⋊5(C2×M4(2)), C20.33(C2×C4), (C2×C4).45D10, C4.Dic5⋊5C2, C22.7(C4×D5), C5⋊2C8⋊11C22, D10.11(C2×C4), (C5×M4(2))⋊3C2, (C2×Dic5).7C4, (C22×D5).5C4, C4.38(C22×D5), C10.28(C22×C4), (C2×C20).25C22, Dic5.13(C2×C4), (C4×D5).38C22, (C2×C4×D5).4C2, C2.16(C2×C4×D5), (C2×C10).25(C2×C4), SmallGroup(160,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×M4(2)
G = < a,b,c,d | a5=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
Subgroups: 184 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, D5×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, D10, C2×M4(2), C4×D5, C22×D5, C2×C4×D5, D5×M4(2)
►On 40 points
(1 26 37 18 10)(2 27 38 19 11)(3 28 39 20 12)(4 29 40 21 13)(5 30 33 22 14)(6 31 34 23 15)(7 32 35 24 16)(8 25 36 17 9)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 37)(34 38)(35 39)(36 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)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
G:=sub<Sym(40)| (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,37)(34,38)(35,39)(36,40), (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)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)>;
G:=Group( (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,37)(34,38)(35,39)(36,40), (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)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40) );
G=PermutationGroup([[(1,26,37,18,10),(2,27,38,19,11),(3,28,39,20,12),(4,29,40,21,13),(5,30,33,22,14),(6,31,34,23,15),(7,32,35,24,16),(8,25,36,17,9)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,37),(34,38),(35,39),(36,40)], [(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),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40)]])
D5×M4(2) is a maximal subgroup of
M4(2)⋊F5 M4(2)⋊3F5 M4(2).F5 M4(2)⋊4F5 M4(2).19D10 M4(2).21D10 C42⋊D10 M4(2).25D10 M4(2)⋊1F5 M4(2)⋊5F5 M4(2).1F5 C40.47C23 C20.72C24 SD16⋊D10 D40⋊C22 C40⋊D6 D15⋊4M4(2)
D5×M4(2) is a maximal quotient of
C40⋊Q8 C42.182D10 C8⋊9D20 D10.6C42 Dic5.14M4(2) Dic5.9M4(2) D10⋊7M4(2) D10⋊4M4(2) Dic5⋊2M4(2) Dic5.5M4(2) C42.200D10 C42.202D10 D10⋊5M4(2) C20⋊5M4(2) Dic5⋊5M4(2) D10⋊8M4(2) C40⋊D4 C40⋊18D4 C40⋊D6 D15⋊4M4(2)
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | D5×M4(2) |
| kernel | D5×M4(2) | C8×D5 | C8⋊D5 | C4.Dic5 | C5×M4(2) | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | M4(2) | D5 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D5×M4(2) ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 33 | 7 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 33 | 1 |
| 0 | 1 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[40,0,0,0,0,40,0,0,0,0,40,33,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;
D5×M4(2) in GAP, Magma, Sage, TeX
D_5\times M_4(2)
% in TeX
G:=Group("D5xM4(2)"); // GroupNames label
G:=SmallGroup(160,127);
// by ID
G=gap.SmallGroup(160,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,50,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=d^2=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=c^5>;
// generators/relations