direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×SD16, C4.7D4, C8⋊3C22, C4.2C23, Q8⋊1C22, D4.1C22, C22.15D4, (C2×C8)⋊5C2, (C2×Q8)⋊3C2, (C2×D4).6C2, C2.12(C2×D4), (C2×C4).27C22, 2-Sylow(GL(3,3)), SmallGroup(32,40)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×SD16
G = < a,b,c | a2=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Character table of C2×SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
►On 16 points - transitive group 16T48
Generators in S16
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)
G:=sub<Sym(16)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)>;
G:=Group( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15) );
G=PermutationGroup([(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15)])
G:=TransitiveGroup(16,48);
Matrix representation of C2×SD16 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 5 | 12 |
| 0 | 5 | 5 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,5,5,0,12,5],[1,0,0,0,1,0,0,0,16] >;
C2×SD16 in GAP, Magma, Sage, TeX
C_2\times {\rm SD}_{16} % in TeX
G:=Group("C2xSD16"); // GroupNames label
G:=SmallGroup(32,40);
// by ID
G=gap.SmallGroup(32,40);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,80,101,483,248,58]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations