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
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, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2.C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C2×C4×D5, C22×F5, D10.3Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8
Character table of D10.3Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 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 | i | 1 | i | i | -i | -i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | 1 | -i | -i | i | i | i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -1 | i | i | -i | -i | i | -1 | 1 | 1 | -i | 1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | 1 | i | -i | i | i | -i | 1 | -1 | -1 | -i | 1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | -1 | -i | -i | i | i | -i | -1 | 1 | 1 | i | 1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | 1 | -i | i | -i | -i | i | 1 | -1 | -1 | i | 1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -1 | -i | -i | i | i | -i | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -1 | i | i | -i | -i | i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | i | -i | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | -i | 1 | -1 | 1 | -1 | i | -i | i | i | 1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | i | 1 | -1 | 1 | -1 | -i | i | -i | -i | 1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | -i | i | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ22 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -√5 | √5 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | √5 | -√5 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -i | -i | i | i | complex lifted from C4×F5 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | i | i | -i | -i | complex lifted from C4×F5 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -√-5 | √-5 | -√-5 | √-5 | complex lifted from C4⋊F5 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | √-5 | -√-5 | √-5 | -√-5 | complex lifted from C4⋊F5 |
►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)
(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 C42⋊4F5 C4×C4⋊F5 C42⋊9F5 C42⋊5F5 C22⋊C4×F5 D10⋊(C4⋊C4) C10.(C4×D4) C4⋊C4×F5 C4⋊C4⋊5F5 C20⋊(C4⋊C4) C4×C22⋊F5 (C22×C4)⋊7F5 D10⋊6(C4⋊C4) (C2×F5)⋊D4 (C2×F5)⋊Q8 D10.20D12 D10.10D12
D10.3Q8 is a maximal quotient of
C42⋊6F5 C42⋊3F5 (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)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 40 | 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 | 2 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 | 0 | 14 |
| 0 | 0 | 0 | 34 | 27 | 14 |
| 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 14 | 0 | 27 | 7 |
| 32 | 23 | 0 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 34 | 27 |
| 0 | 0 | 0 | 27 | 7 | 34 |
| 0 | 0 | 14 | 34 | 7 | 27 |
| 0 | 0 | 14 | 27 | 34 | 0 |
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
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