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G = D10.3Q8order 160 = 25·5

3rd non-split extension by D10 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.3Q8, D10.10D4, C10.5C42, (C2×F5)⋊C4, (C2×C4)⋊2F5, (C2×C20)⋊2C4, D5.(C4⋊C4), C2.5(C4×F5), C2.3(C4⋊F5), C10.7(C4⋊C4), (C2×Dic5)⋊6C4, D10.7(C2×C4), C5⋊(C2.C42), D5.(C22⋊C4), C2.2(C22⋊F5), (C22×F5).1C2, C22.13(C2×F5), C10.4(C22⋊C4), (C22×D5).35C22, (C2×C4×D5).9C2, (C2×C10).9(C2×C4), SmallGroup(160,81)

Series: Derived Chief Lower central Upper central

C1C10 — D10.3Q8
C1C5D5D10C22×D5C22×F5 — D10.3Q8
C5C10 — D10.3Q8
C1C22C2×C4

Generators and relations for D10.3Q8
 G = < a,b,c,d | a10=b2=c4=1, d2=a4bc2, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=a5c-1 >

Subgroups: 268 in 76 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×6], C5, C2×C4, C2×C4 [×11], C23, D5 [×4], C10 [×3], C22×C4 [×3], Dic5, C20, F5 [×4], D10 [×2], D10 [×4], C2×C10, C2.C42, C4×D5 [×2], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C2×C4×D5, C22×F5 [×2], D10.3Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C2×F5, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8

Character table of D10.3Q8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L510A10B10C20A20B20C20D
 size 11115555221010101010101010101044444444
ρ11111111111111111111111111111    trivial
ρ21111111111-11-1-1-1-1-1-1-1111111111    linear of order 2
ρ311111111-1-11-1-1-1-1-1111-11111-1-1-1-1    linear of order 2
ρ411111111-1-1-1-11111-1-1-1-11111-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-1-1i1ii-i-i-i-ii11111-1-1-1-1    linear of order 4
ρ61111-1-1-1-1-1-1-i1-i-iiiii-i11111-1-1-1-1    linear of order 4
ρ711-1-1-1-111i-i-1ii-i-ii-111-i1-11-1-i-iii    linear of order 4
ρ811-1-1-1-111i-i1i-iii-i1-1-1-i1-11-1-i-iii    linear of order 4
ρ911-1-1-1-111-ii-1-i-iii-i-111i1-11-1ii-i-i    linear of order 4
ρ1011-1-1-1-111-ii1-ii-i-ii1-1-1i1-11-1ii-i-i    linear of order 4
ρ111111-1-1-1-111i-1-i-iii-i-ii-111111111    linear of order 4
ρ121111-1-1-1-111-i-1ii-i-iii-i-111111111    linear of order 4
ρ1311-1-111-1-1i-ii-i-11-11-ii-ii1-11-1-i-iii    linear of order 4
ρ1411-1-111-1-1i-i-i-i1-11-1i-iii1-11-1-i-iii    linear of order 4
ρ1511-1-111-1-1-iiii1-11-1-ii-i-i1-11-1ii-i-i    linear of order 4
ρ1611-1-111-1-1-ii-ii-11-11i-ii-i1-11-1ii-i-i    linear of order 4
ρ172-22-22-2-220000000000002-2-220000    orthogonal lifted from D4
ρ182-22-2-222-20000000000002-2-220000    orthogonal lifted from D4
ρ192-2-22-22-2200000000000022-2-20000    orthogonal lifted from D4
ρ202-2-222-22-200000000000022-2-20000    symplectic lifted from Q8, Schur index 2
ρ2144440000440000000000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2244440000-4-40000000000-1-1-1-11111    orthogonal lifted from C2×F5
ρ234-44-40000000000000000-111-1-555-5    orthogonal lifted from C22⋊F5
ρ244-44-40000000000000000-111-15-5-55    orthogonal lifted from C22⋊F5
ρ2544-4-40000-4i4i0000000000-11-11-i-iii    complex lifted from C4×F5
ρ2644-4-400004i-4i0000000000-11-11ii-i-i    complex lifted from C4×F5
ρ274-4-440000000000000000-1-111--5-5--5-5    complex lifted from C4⋊F5
ρ284-4-440000000000000000-1-111-5--5-5--5    complex lifted from C4⋊F5

Smallest permutation representation of D10.3Q8
On 40 points
Generators in S40
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 32)(12 31)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)
(1 33 30 11)(2 34 21 12)(3 35 22 13)(4 36 23 14)(5 37 24 15)(6 38 25 16)(7 39 26 17)(8 40 27 18)(9 31 28 19)(10 32 29 20)
(1 33 6 38)(2 40 5 31)(3 37 4 34)(7 35 10 36)(8 32 9 39)(11 25 16 30)(12 22 15 23)(13 29 14 26)(17 27 20 28)(18 24 19 21)

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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,32)(12,31)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33), (1,33,30,11)(2,34,21,12)(3,35,22,13)(4,36,23,14)(5,37,24,15)(6,38,25,16)(7,39,26,17)(8,40,27,18)(9,31,28,19)(10,32,29,20), (1,33,6,38)(2,40,5,31)(3,37,4,34)(7,35,10,36)(8,32,9,39)(11,25,16,30)(12,22,15,23)(13,29,14,26)(17,27,20,28)(18,24,19,21)>;

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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,32)(12,31)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33), (1,33,30,11)(2,34,21,12)(3,35,22,13)(4,36,23,14)(5,37,24,15)(6,38,25,16)(7,39,26,17)(8,40,27,18)(9,31,28,19)(10,32,29,20), (1,33,6,38)(2,40,5,31)(3,37,4,34)(7,35,10,36)(8,32,9,39)(11,25,16,30)(12,22,15,23)(13,29,14,26)(17,27,20,28)(18,24,19,21) );

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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,32),(12,31),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33)], [(1,33,30,11),(2,34,21,12),(3,35,22,13),(4,36,23,14),(5,37,24,15),(6,38,25,16),(7,39,26,17),(8,40,27,18),(9,31,28,19),(10,32,29,20)], [(1,33,6,38),(2,40,5,31),(3,37,4,34),(7,35,10,36),(8,32,9,39),(11,25,16,30),(12,22,15,23),(13,29,14,26),(17,27,20,28),(18,24,19,21)])

D10.3Q8 is a maximal subgroup of
D10.1D8  D10.1Q16  D10.SD16  D10.Q16  C424F5  C4×C4⋊F5  C429F5  C425F5  C22⋊C4×F5  D10⋊(C4⋊C4)  C10.(C4×D4)  C4⋊C4×F5  C4⋊C45F5  C20⋊(C4⋊C4)  C4×C22⋊F5  (C22×C4)⋊7F5  D106(C4⋊C4)  (C2×F5)⋊D4  (C2×F5)⋊Q8  D10.20D12  D10.10D12
D10.3Q8 is a maximal quotient of
C426F5  C423F5  (C22×F5)⋊C4  C22⋊C4.F5  D10.18D8  C20.C42  D10.3M4(2)  D10.10D8  (C2×C8)⋊F5  C20.24C42  C20.10C42  C20.25C42  M4(2)⋊F5  M4(2)⋊3F5  M4(2).F5  M4(2)⋊4F5  C22⋊F5⋊C4  C10.(C4⋊C8)  C22.F5⋊C4  D10.20D12  D10.10D12

Matrix representation of D10.3Q8 in GL6(𝔽41)

4000000
0400000
0000401
0000400
0010400
0001400
,
100000
010000
0000040
0000400
0004000
0040000
,
120000
0400000
00727014
000342714
001427340
00140277
,
32230000
990000
00703427
00027734
001434727
001427340

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[32,9,0,0,0,0,23,9,0,0,0,0,0,0,7,0,14,14,0,0,0,27,34,27,0,0,34,7,7,34,0,0,27,34,27,0] >;

D10.3Q8 in GAP, Magma, Sage, TeX

D_{10}._3Q_8
% in TeX

G:=Group("D10.3Q8");
// GroupNames label

G:=SmallGroup(160,81);
// by ID

G=gap.SmallGroup(160,81);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,2309,1169]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^4*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

Export

Character table of D10.3Q8 in TeX

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