p-group, metabelian, nilpotent (class 4), monomial
Aliases: (C2×C4).1Q16, (C2×C4).5SD16, C22.19C4≀C2, (C22×C4).33D4, C2.C42.5C4, C22.C42.8C2, C22.60(C23⋊C4), C2.5(C42.C4), C2.7(C23.D4), C23.160(C22⋊C4), C22.18(Q8⋊C4), C2.6(C23.31D4), C23.83C23.1C2, C22.M4(2).5C2, (C2×C4⋊C4).9C4, (C2×C4⋊C4).7C22, (C22×C4).7(C2×C4), SmallGroup(128,85)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
C1 — C22 — C23 — C2×C4⋊C4 — (C2×C4).Q16 |
C1 — C22 — C23 — C2×C4⋊C4 — (C2×C4).Q16 |
Generators and relations for (C2×C4).Q16
G = < a,b,c,d | a2=b4=c8=1, d2=ab2c4, cbc-1=ab=ba, cac-1=ab2, ad=da, dbd-1=ab-1, dcd-1=b-1c-1 >
Character table of (C2×C4).Q16
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 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 | 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 | -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 | -1 | 1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | i | -i | -i | -i | i | -i | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | i | -i | -i | -i | i | i | i | -i | linear of order 4 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | -i | i | i | i | -i | i | linear of order 4 |
ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | 0 | 0 | √2 | 0 | 0 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | 0 | 0 | -√2 | 0 | 0 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
ρ13 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1+i | -1-i | 0 | 1-i | -1+i | 0 | 0 | complex lifted from C4≀C2 |
ρ14 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1-i | -1+i | 0 | 1+i | -1-i | 0 | 0 | complex lifted from C4≀C2 |
ρ15 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -1+i | 1-i | 0 | -1-i | 1+i | 0 | 0 | complex lifted from C4≀C2 |
ρ16 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | 0 | 0 | √-2 | 0 | 0 | √-2 | -√-2 | complex lifted from SD16 |
ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | 0 | 0 | -√-2 | 0 | 0 | -√-2 | √-2 | complex lifted from SD16 |
ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -1-i | 1+i | 0 | -1+i | 1-i | 0 | 0 | complex lifted from C4≀C2 |
ρ19 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ20 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
(1 29)(2 6)(3 31)(4 8)(5 25)(7 27)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)(26 30)(28 32)
(1 13 25 18)(2 23 26 10)(3 20 27 15)(4 12 28 17)(5 9 29 22)(6 19 30 14)(7 24 31 11)(8 16 32 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(2 12 26 17)(3 27)(4 14 28 19)(6 16 30 21)(7 31)(8 10 32 23)(9 13)(11 20)(15 24)(18 22)
G:=sub<Sym(32)| (1,29)(2,6)(3,31)(4,8)(5,25)(7,27)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23)(26,30)(28,32), (1,13,25,18)(2,23,26,10)(3,20,27,15)(4,12,28,17)(5,9,29,22)(6,19,30,14)(7,24,31,11)(8,16,32,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,12,26,17)(3,27)(4,14,28,19)(6,16,30,21)(7,31)(8,10,32,23)(9,13)(11,20)(15,24)(18,22)>;
G:=Group( (1,29)(2,6)(3,31)(4,8)(5,25)(7,27)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23)(26,30)(28,32), (1,13,25,18)(2,23,26,10)(3,20,27,15)(4,12,28,17)(5,9,29,22)(6,19,30,14)(7,24,31,11)(8,16,32,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,12,26,17)(3,27)(4,14,28,19)(6,16,30,21)(7,31)(8,10,32,23)(9,13)(11,20)(15,24)(18,22) );
G=PermutationGroup([(1,29),(2,6),(3,31),(4,8),(5,25),(7,27),(9,18),(10,14),(11,20),(12,16),(13,22),(15,24),(17,21),(19,23),(26,30),(28,32)], [(1,13,25,18),(2,23,26,10),(3,20,27,15),(4,12,28,17),(5,9,29,22),(6,19,30,14),(7,24,31,11),(8,16,32,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(2,12,26,17),(3,27),(4,14,28,19),(6,16,30,21),(7,31),(8,10,32,23),(9,13),(11,20),(15,24),(18,22)])
Matrix representation of (C2×C4).Q16 ►in GL6(𝔽17)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 12 | 0 | 1 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 13 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 13 | 0 |
0 | 0 | 0 | 0 | 4 | 4 |
7 | 13 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 12 | 1 | 2 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 15 | 11 | 11 | 5 |
10 | 4 | 0 | 0 | 0 | 0 |
13 | 7 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 12 | 1 | 2 |
0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,4,0,0,0,0,0,4],[7,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,15,0,0,0,12,4,11,0,0,1,1,0,11,0,0,0,2,0,5],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,0,0,0,0,0,16,12,0,0,0,0,0,1,16,0,0,0,0,2,16] >;
(C2×C4).Q16 in GAP, Magma, Sage, TeX
(C_2\times C_4).Q_{16}
% in TeX
G:=Group("(C2xC4).Q16");
// GroupNames label
G:=SmallGroup(128,85);
// by ID
G=gap.SmallGroup(128,85);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,1690,521,248,2804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=a*b^2*c^4,c*b*c^-1=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,d*b*d^-1=a*b^-1,d*c*d^-1=b^-1*c^-1>;
// generators/relations
Export
Subgroup lattice of (C2×C4).Q16 in TeX
Character table of (C2×C4).Q16 in TeX