metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5⋊M4(2), Dic5.11C23, D5⋊C8⋊4C2, C5⋊C8⋊2C22, C4.F5⋊5C2, (C2×C4).8F5, (C2×C20).8C4, (C4×D5).8C4, C5⋊2(C2×M4(2)), C4.21(C2×F5), C20.20(C2×C4), C22.F5⋊3C2, C22.6(C2×F5), C2.5(C22×F5), D10.14(C2×C4), C10.3(C22×C4), (C22×D5).9C4, Dic5.16(C2×C4), (C4×D5).34C22, (C2×Dic5).56C22, (C2×C4×D5).15C2, (C2×C10).15(C2×C4), SmallGroup(160,202)
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, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=c5 >
Subgroups: 196 in 68 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, D5⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, F5, C2×M4(2), C2×F5, C22×F5, D5⋊M4(2)
Character table of D5⋊M4(2)
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | i | -i | -i | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | -i | i | i | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | i | -i | -i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | -i | i | i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | -i | i | -i | i | i | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | i | -i | i | -i | -i | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ17 | 2 | -2 | 0 | -2 | 2 | 0 | 2i | -2i | 0 | -2i | 2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | 0 | 2 | -2 | 0 | -2i | 2i | 0 | -2i | 2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | 0 | -2 | 2 | 0 | -2i | 2i | 0 | 2i | -2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | 0 | 2 | -2 | 0 | 2i | -2i | 0 | 2i | -2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | i | -i | -√-5 | √-5 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | i | -i | √-5 | -√-5 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | -i | i | √-5 | -√-5 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | -i | i | -√-5 | √-5 | complex faithful |
►On 40 points
(1 26 37 19 10)(2 20 27 11 38)(3 12 21 39 28)(4 40 13 29 22)(5 30 33 23 14)(6 24 31 15 34)(7 16 17 35 32)(8 36 9 25 18)
(1 10)(2 38)(3 28)(4 22)(5 14)(6 34)(7 32)(8 18)(11 20)(12 39)(15 24)(16 35)(19 26)(23 30)(25 36)(29 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)
G:=sub<Sym(40)| (1,26,37,19,10)(2,20,27,11,38)(3,12,21,39,28)(4,40,13,29,22)(5,30,33,23,14)(6,24,31,15,34)(7,16,17,35,32)(8,36,9,25,18), (1,10)(2,38)(3,28)(4,22)(5,14)(6,34)(7,32)(8,18)(11,20)(12,39)(15,24)(16,35)(19,26)(23,30)(25,36)(29,40), (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)>;
G:=Group( (1,26,37,19,10)(2,20,27,11,38)(3,12,21,39,28)(4,40,13,29,22)(5,30,33,23,14)(6,24,31,15,34)(7,16,17,35,32)(8,36,9,25,18), (1,10)(2,38)(3,28)(4,22)(5,14)(6,34)(7,32)(8,18)(11,20)(12,39)(15,24)(16,35)(19,26)(23,30)(25,36)(29,40), (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) );
G=PermutationGroup([[(1,26,37,19,10),(2,20,27,11,38),(3,12,21,39,28),(4,40,13,29,22),(5,30,33,23,14),(6,24,31,15,34),(7,16,17,35,32),(8,36,9,25,18)], [(1,10),(2,38),(3,28),(4,22),(5,14),(6,34),(7,32),(8,18),(11,20),(12,39),(15,24),(16,35),(19,26),(23,30),(25,36),(29,40)], [(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)]])
D5⋊M4(2) is a maximal subgroup of
C42⋊6F5 C42⋊3F5 (C2×C8)⋊F5 C20.25C42 M4(2)⋊F5 M4(2).F5 C20.12C42 (C8×D5).C4 M4(2)×F5 M4(2).1F5 (C4×D5).D4 (D4×C10)⋊C4 (C2×D4)⋊6F5 D5⋊(C4.D4) (C2×Q8)⋊4F5 (C2×Q8)⋊6F5 (C2×Q8).7F5 D5⋊C4≀C2 C4○D4⋊F5 Dic5.C24 Dic5.20C24 Dic5.21C24 Dic5.22C24 D15⋊M4(2) C5⋊C8⋊D6 D15⋊2M4(2) C60.59(C2×C4)
D5⋊M4(2) is a maximal quotient of
C42.5F5 C4×C4.F5 C42.6F5 C42.11F5 C42.14F5 C42.15F5 C42.7F5 D10⋊M4(2) Dic5⋊M4(2) D10⋊2M4(2) Dic5.M4(2) Dic5.12M4(2) C4×C22.F5 D10.11M4(2) D10⋊9M4(2) D10⋊10M4(2) Dic5.13M4(2) C20⋊8M4(2) D15⋊M4(2) C5⋊C8⋊D6 D15⋊2M4(2) C60.59(C2×C4)
Matrix representation of D5⋊M4(2) ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 40 | 6 |
| 0 | 0 | 35 | 35 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 6 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 |
| 28 | 9 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,40,35,0,0,6,35],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,6,1],[0,0,32,28,0,0,0,9,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40] >;
D5⋊M4(2) in GAP, Magma, Sage, TeX
D_5\rtimes M_4(2)
% in TeX
G:=Group("D5:M4(2)"); // GroupNames label
G:=SmallGroup(160,202);
// by ID
G=gap.SmallGroup(160,202);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,69,2309,599]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^5>;
// generators/relations
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