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G = D5⋊M4(2)  order 160 = 25·5

The semidirect product of D5 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5⋊M4(2), Dic5.11C23, D5⋊C84C2, C5⋊C82C22, C4.F55C2, (C2×C4).8F5, (C2×C20).8C4, (C4×D5).8C4, C52(C2×M4(2)), C4.21(C2×F5), C20.20(C2×C4), C22.F53C2, 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

C1C10 — D5⋊M4(2)
C1C5C10Dic5C5⋊C8D5⋊C8 — D5⋊M4(2)
C5C10 — D5⋊M4(2)
C1C4C2×C4

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 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×M4(2), C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×2], C2×C4×D5, D5⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, F5, C2×M4(2), C2×F5 [×3], C22×F5, D5⋊M4(2)

Character table of D5⋊M4(2)

 class 12A2B2C2D2E4A4B4C4D4E4F58A8B8C8D8E8F8G8H10A10B10C20A20B20C20D
 size 11255101125510410101010101010104444444
ρ11111111111111111111111111111    trivial
ρ2111-1-1-1-1-1-11111-11111-1-1-1111-1-1-1-1    linear of order 2
ρ311-111-111-111-111-1-111-1-11-1-1111-1-1    linear of order 2
ρ411-1-1-11-1-1111-11-1-1-11111-1-1-11-1-111    linear of order 2
ρ5111-1-1-1-1-1-111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ61111111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ711-111-111-111-11-111-1-111-1-1-1111-1-1    linear of order 2
ρ811-1-1-11-1-1111-11111-1-1-1-11-1-11-1-111    linear of order 2
ρ911-1-1-1111-1-1-111ii-i-iii-i-i-1-1111-1-1    linear of order 4
ρ1011-1-1-1111-1-1-111-i-iii-i-iii-1-1111-1-1    linear of order 4
ρ11111111-1-1-1-1-1-11ii-ii-i-ii-i111-1-1-1-1    linear of order 4
ρ12111111-1-1-1-1-1-11-i-ii-iii-ii111-1-1-1-1    linear of order 4
ρ1311-111-1-1-11-1-111-ii-i-ii-iii-1-11-1-111    linear of order 4
ρ1411-111-1-1-11-1-111i-iii-ii-i-i-1-11-1-111    linear of order 4
ρ15111-1-1-1111-1-1-11-ii-ii-ii-ii1111111    linear of order 4
ρ16111-1-1-1111-1-1-11i-ii-ii-ii-i1111111    linear of order 4
ρ172-20-2202i-2i0-2i2i020000000000-22i-2i00    complex lifted from M4(2)
ρ182-202-20-2i2i0-2i2i020000000000-2-2i2i00    complex lifted from M4(2)
ρ192-20-220-2i2i02i-2i020000000000-2-2i2i00    complex lifted from M4(2)
ρ202-202-202i-2i02i-2i020000000000-22i-2i00    complex lifted from M4(2)
ρ2144-400044-4000-10000000011-1-1-111    orthogonal lifted from C2×F5
ρ22444000-4-4-4000-100000000-1-1-11111    orthogonal lifted from C2×F5
ρ2344-4000-4-44000-10000000011-111-1-1    orthogonal lifted from C2×F5
ρ24444000444000-100000000-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ254-40000-4i4i0000-100000000-551i-i--5-5    complex faithful
ρ264-40000-4i4i0000-1000000005-51i-i-5--5    complex faithful
ρ274-400004i-4i0000-100000000-551-ii-5--5    complex faithful
ρ284-400004i-4i0000-1000000005-51-ii--5-5    complex faithful

Smallest permutation representation of D5⋊M4(2)
On 40 points
Generators in S40
(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
C426F5  C423F5  (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  D152M4(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)  D102M4(2)  Dic5.M4(2)  Dic5.12M4(2)  C4×C22.F5  D10.11M4(2)  D109M4(2)  D1010M4(2)  Dic5.13M4(2)  C208M4(2)  D15⋊M4(2)  C5⋊C8⋊D6  D152M4(2)  C60.59(C2×C4)

Matrix representation of D5⋊M4(2) in GL4(𝔽41) generated by

0100
40600
00406
003535
,
0100
1000
00406
0001
,
0010
0001
32000
28900
,
1000
0100
00400
00040
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

Export

Character table of D5⋊M4(2) in TeX

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