p-group, non-abelian, nilpotent (class 4), monomial
Aliases: C42.17D4, 2- 1+4.2C22, (C2×D4).38D4, (C2×Q8).37D4, C2.28C2≀C22, D4.8D4⋊2C2, C4⋊Q8.98C22, D4.10D4⋊2C2, (C2×Q8).3C23, C42.3C4⋊2C2, C42.C4⋊3C2, C22.52C22≀C2, C4.10D4.2C22, C4.4D4.23C22, C22.49C24⋊2C2, (C2×C4).21(C2×D4), SmallGroup(128,936)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.17D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1b, cbc-1=a2b, bd=db, dcd=b2c3 >
Subgroups: 304 in 118 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, M4(2), D8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4.10D4, C4.10D4, C4≀C2, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4.4D4, C4⋊Q8, C8⋊C22, C8.C22, 2- 1+4, C42.C4, C42.3C4, D4.8D4, D4.10D4, C22.49C24, C42.17D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, C42.17D4
Character table of C42.17D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 | 16 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C22 |
| ρ16 | 4 | 4 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C22 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2i | -2i | -2 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | 2i | 2i | -2 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T337
(1 9 5 13)(3 15 7 11)
(1 9 5 13)(2 14 6 10)(3 15 7 11)(4 12 8 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 6)(2 5)(3 4)(7 8)(9 10)(11 16)(12 15)(13 14)
G:=sub<Sym(16)| (1,9,5,13)(3,15,7,11), (1,9,5,13)(2,14,6,10)(3,15,7,11)(4,12,8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,6)(2,5)(3,4)(7,8)(9,10)(11,16)(12,15)(13,14)>;
G:=Group( (1,9,5,13)(3,15,7,11), (1,9,5,13)(2,14,6,10)(3,15,7,11)(4,12,8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,6)(2,5)(3,4)(7,8)(9,10)(11,16)(12,15)(13,14) );
G=PermutationGroup([[(1,9,5,13),(3,15,7,11)], [(1,9,5,13),(2,14,6,10),(3,15,7,11),(4,12,8,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,6),(2,5),(3,4),(7,8),(9,10),(11,16),(12,15),(13,14)]])
G:=TransitiveGroup(16,337);
►On 16 points - transitive group 16T407
(1 14 5 10)(3 12 7 16)
(1 14 5 10)(2 11 6 15)(3 12 7 16)(4 9 8 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)
G:=sub<Sym(16)| (1,14,5,10)(3,12,7,16), (1,14,5,10)(2,11,6,15)(3,12,7,16)(4,9,8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)>;
G:=Group( (1,14,5,10)(3,12,7,16), (1,14,5,10)(2,11,6,15)(3,12,7,16)(4,9,8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14) );
G=PermutationGroup([[(1,14,5,10),(3,12,7,16)], [(1,14,5,10),(2,11,6,15),(3,12,7,16),(4,9,8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]])
G:=TransitiveGroup(16,407);
Matrix representation of C42.17D4 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 4 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 2 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 3 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,1,0,4,0,0,0,0,4,0,4,0,0,0],[0,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0],[0,1,0,0,0,0,0,3,4,0,0,0,0,0,2,0],[0,4,0,0,4,0,0,0,0,0,0,3,0,0,2,0] >;
C42.17D4 in GAP, Magma, Sage, TeX
C_4^2._{17}D_4 % in TeX
G:=Group("C4^2.17D4"); // GroupNames label
G:=SmallGroup(128,936);
// by ID
G=gap.SmallGroup(128,936);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,723,2019,1018,297,248,2804,1971,718,375,172,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations
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