p-group, non-abelian, nilpotent (class 4), monomial
Aliases: C42.13D4, 2+ 1+4.2C22, (C2×D4).35D4, C22⋊C4.1D4, C42⋊C4⋊6C2, C2.22C2≀C22, D4⋊4D4.2C2, (C2×D4).3C23, (C22×C4).26D4, C23.15(C2×D4), C23.D4⋊2C2, C23.7D4⋊2C2, C23⋊C4.2C22, C22.46C22≀C2, C4⋊1D4.56C22, C4.D4.2C22, C22.53C24⋊2C2, C22.D4.4C22, (C2×C4).15(C2×D4), SmallGroup(128,930)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.13D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b, cbc-1=a2b, bd=db, dcd=b2c-1 >
Subgroups: 336 in 123 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C23⋊C4, C4.D4, C4≀C2, C4×D4, C4×Q8, C22.D4, C22.D4, C4.4D4, C4⋊1D4, C8⋊C22, 2+ 1+4, C23.D4, C42⋊C4, D4⋊4D4, C23.7D4, C22.53C24, C42.13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, C42.13D4
Character table of C42.13D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8 | |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C22 |
| ρ16 | 4 | 4 | -4 | 0 | 0 | 2 | 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 | 0 | -2 | 2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T344
(9 10 11 12)(13 14 15 16)
(1 3 4 2)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 4 14)(2 15)(3 13)(5 10 7 12)(6 11)(8 9)
(1 12)(2 11)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)
G:=sub<Sym(16)| (9,10,11,12)(13,14,15,16), (1,3,4,2)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,4,14)(2,15)(3,13)(5,10,7,12)(6,11)(8,9), (1,12)(2,11)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)>;
G:=Group( (9,10,11,12)(13,14,15,16), (1,3,4,2)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,4,14)(2,15)(3,13)(5,10,7,12)(6,11)(8,9), (1,12)(2,11)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15) );
G=PermutationGroup([[(9,10,11,12),(13,14,15,16)], [(1,3,4,2),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,4,14),(2,15),(3,13),(5,10,7,12),(6,11),(8,9)], [(1,12),(2,11),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15)]])
G:=TransitiveGroup(16,344);
►On 16 points - transitive group 16T380
(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 12 11 10)(13 16 15 14)
(1 16 6 12)(2 14 5 10)(3 9 8 15)(4 11 7 13)
(1 13)(2 15)(3 12)(4 10)(5 9)(6 11)(7 14)(8 16)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,2,7)(3,6,4,5)(9,12,11,10)(13,16,15,14), (1,16,6,12)(2,14,5,10)(3,9,8,15)(4,11,7,13), (1,13)(2,15)(3,12)(4,10)(5,9)(6,11)(7,14)(8,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,2,7)(3,6,4,5)(9,12,11,10)(13,16,15,14), (1,16,6,12)(2,14,5,10)(3,9,8,15)(4,11,7,13), (1,13)(2,15)(3,12)(4,10)(5,9)(6,11)(7,14)(8,16) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,8,2,7),(3,6,4,5),(9,12,11,10),(13,16,15,14)], [(1,16,6,12),(2,14,5,10),(3,9,8,15),(4,11,7,13)], [(1,13),(2,15),(3,12),(4,10),(5,9),(6,11),(7,14),(8,16)]])
G:=TransitiveGroup(16,380);
Matrix representation of C42.13D4 ►in GL4(𝔽5) generated by
| 0 | 2 | 1 | 2 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 2 | 1 | 0 | 1 |
| 4 | 0 | 0 | 4 |
| 2 | 0 | 2 | 1 |
| 0 | 2 | 0 | 3 |
| 2 | 0 | 0 | 1 |
| 2 | 0 | 0 | 2 |
| 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 3 |
| 1 | 0 | 2 | 0 |
| 0 | 0 | 3 | 1 |
| 0 | 0 | 4 | 0 |
| 0 | 1 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,2,2,1,0,1,1,0,0,0,2,0,3,1],[4,2,0,2,0,0,2,0,0,2,0,0,4,1,3,1],[2,0,0,0,0,0,0,2,0,4,0,0,2,3,4,3],[1,0,0,0,0,0,0,1,2,3,4,3,0,1,0,0] >;
C42.13D4 in GAP, Magma, Sage, TeX
C_4^2._{13}D_4 % in TeX
G:=Group("C4^2.13D4"); // GroupNames label
G:=SmallGroup(128,930);
// by ID
G=gap.SmallGroup(128,930);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,352,297,1971,375,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,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^-1>;
// generators/relations
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