p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.31D4, C22.2Q16, C22.2SD16, C4⋊C4⋊2C4, C2.5C4≀C2, (C2×Q8)⋊1C4, (C2×C4).95D4, C22⋊C8.2C2, C22⋊Q8.1C2, C2.5(C23⋊C4), C2.3(Q8⋊C4), C2.C42.4C2, (C22×C4).17C22, C22.36(C22⋊C4), (C2×C4).9(C2×C4), SmallGroup(64,9)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.31D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd3 >
Character table of C23.31D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | 1 | i | -i | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | 1 | -i | i | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | 1 | -i | i | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | 1 | i | -i | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 1-i | 1+i | 0 | -1-i | -1+i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | -1+i | -1-i | 0 | 1+i | 1-i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 1+i | 1-i | 0 | -1+i | -1-i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | -1-i | -1+i | 0 | 1-i | 1+i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
►On 16 points - transitive group 16T161
(2 15)(4 9)(6 11)(8 13)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4 15 9)(3 12)(6 8 11 13)(7 16)
G:=sub<Sym(16)| (2,15)(4,9)(6,11)(8,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4,15,9)(3,12)(6,8,11,13)(7,16)>;
G:=Group( (2,15)(4,9)(6,11)(8,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4,15,9)(3,12)(6,8,11,13)(7,16) );
G=PermutationGroup([[(2,15),(4,9),(6,11),(8,13)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,4,15,9),(3,12),(6,8,11,13),(7,16)]])
G:=TransitiveGroup(16,161);
C23.31D4 is a maximal subgroup of
C24.53D4 C24.150D4 C24.55D4 C24.57D4 C42.58D4 C24.58D4 C42.60D4 C24.59D4 C42.62D4 C24.61D4 C42.63D4 C23⋊SD16 C4⋊C4.D4 (C2×C4)⋊SD16 C24.9D4 C23⋊Q16 C4⋊C4.6D4 (C2×C4)⋊Q16 C24.12D4 C24.14D4 C4⋊C4.12D4 (C2×C4).SD16 C24.15D4 C24.17D4 C4⋊C4.18D4 C4⋊C4.20D4 C24.18D4 C23.7S4
C2p.C4≀C2: C42.375D4 C42.404D4 C42.56D4 C42.57D4 C4⋊Dic3⋊C4 C23.35D12 (C6×Q8)⋊C4 C4⋊Dic5⋊C4 ...
C23.31D4 is a maximal quotient of
C23.Q16 C24.4D4 D10.1Q16 D10.Q16
C23.D4p: C23.30D8 C23.35D12 C23.30D20 C23.30D28 ...
(C2×C2p).Q16: C4⋊C4⋊C8 (C2×Q8)⋊C8 (C2×C4).Q16 C2.7C2≀C4 C4⋊Dic3⋊C4 (C6×Q8)⋊C4 C4⋊Dic5⋊C4 C10.29C4≀C2 ...
Matrix representation of C23.31D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 9 | 9 |
| 0 | 0 | 16 | 8 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 14 | 13 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,15,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,9,16,0,0,9,8],[13,0,0,0,0,4,0,0,0,0,16,14,0,0,0,13] >;
C23.31D4 in GAP, Magma, Sage, TeX
C_2^3._{31}D_4 % in TeX
G:=Group("C2^3.31D4"); // GroupNames label
G:=SmallGroup(64,9);
// by ID
G=gap.SmallGroup(64,9);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*c*d^3>;
// generators/relations
Export
Subgroup lattice of C23.31D4 in TeX
Character table of C23.31D4 in TeX