p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22.2D8, C23.30D4, C22.7SD16, C4⋊C4⋊1C4, C2.4C4≀C2, (C2×D4)⋊1C4, C22⋊C8⋊2C2, (C2×C4).94D4, C4⋊D4.1C2, C2.4(C23⋊C4), C2.3(D4⋊C4), C2.C42⋊6C2, (C22×C4).16C22, C22.35(C22⋊C4), (C2×C4).8(C2×C4), SmallGroup(64,8)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.SD16
G = < a,b,c,d | a2=b2=c8=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=abc3 >
Character table of C22.SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | -1 | -i | i | 1 | i | -i | 1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | i | 1 | i | -i | -1 | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -i | 1 | -i | i | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | i | -i | 1 | -i | i | 1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | 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 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | -1+i | -1-i | 0 | 1+i | 1-i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 1+i | 1-i | 0 | -1+i | -1-i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | -1-i | -1+i | 0 | 1-i | 1+i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 1-i | 1+i | 0 | -1-i | -1+i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| 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 16T163
(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 6)(2 5)(3 16)(4 15)(7 12)(8 11)(9 14)(10 13)
G:=sub<Sym(16)| (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,6)(2,5)(3,16)(4,15)(7,12)(8,11)(9,14)(10,13)>;
G:=Group( (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,6)(2,5)(3,16)(4,15)(7,12)(8,11)(9,14)(10,13) );
G=PermutationGroup([[(1,5),(2,10),(3,7),(4,12),(6,14),(8,16),(9,13),(11,15)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,6),(2,5),(3,16),(4,15),(7,12),(8,11),(9,14),(10,13)]])
G:=TransitiveGroup(16,163);
C22.SD16 is a maximal subgroup of
C42.375D4 C24.53D4 C24.150D4 C42.55D4 C24.54D4 C42.57D4 C24.56D4 C42.58D4 C24.58D4 C42.59D4 C24.59D4 C42.63D4 C23⋊D8 C4⋊C4.D4 (C2×C4)⋊D8 C24.9D4 C23⋊2SD16 C4⋊C4.6D4 Q8⋊D4⋊C2 C24.12D4 C4⋊C4.12D4 C24.15D4 C24.16D4 C4⋊C4.18D4 C4⋊C4.19D4 C24.18D4 C23.8S4 D10.1D8 D10.SD16
C23.D4p: C24.60D4 C23.5D8 C22.2D24 C22.2D40 C22.2D56 ...
(C2×C2p).D8: C42.403D4 C42.61D4 (C2×C4).5D8 C6.C4≀C2 (C6×D4)⋊C4 (C2×D20)⋊C4 C4⋊C4⋊Dic5 C14.C4≀C2 ...
C22.SD16 is a maximal quotient of
(C2×C4).98D8 C4⋊C4⋊C8 C23.30D8 C24.D4 C23.4D8 C2.C2≀C4 (C2×C4).D8
C22.D8p: C22.SD32 C22.2D24 C22.2D40 C22.2D56 ...
C2p.C4≀C2: C23.D8 C23.2D8 C23.SD16 C23.2SD16 C23.32D8 C23.12SD16 C23.13SD16 C6.C4≀C2 ...
Matrix representation of C22.SD16 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 9 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 2 | 8 |
| 0 | 0 | 0 | 15 |
| 14 | 14 | 0 | 0 |
| 14 | 3 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 11 | 8 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,9,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,2,0,0,0,8,15],[14,14,0,0,14,3,0,0,0,0,9,11,0,0,2,8] >;
C22.SD16 in GAP, Magma, Sage, TeX
C_2^2.{\rm SD}_{16} % in TeX
G:=Group("C2^2.SD16"); // GroupNames label
G:=SmallGroup(64,8);
// by ID
G=gap.SmallGroup(64,8);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=a*b*c^3>;
// generators/relations
Export
Subgroup lattice of C22.SD16 in TeX
Character table of C22.SD16 in TeX