p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8:4C4, Q16:4C4, C8.26D4, SD16:2C4, C42.12C22, M4(2).12C22, C4wrC2:6C2, C8oD4:7C2, C8.6(C2xC4), C8:C4:3C2, C4oD8.3C2, D4.4(C2xC4), C2.19(C4xD4), C4.80(C2xD4), Q8.4(C2xC4), C8.C4:4C2, (C2xC4).80C23, (C2xC8).51C22, C4.16(C22xC4), C4oD4.8C22, C22.2(C4oD4), SmallGroup(64,125)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.26D4
G = < a,b,c | a8=b4=1, c2=a2, bab-1=cac-1=a5, cbc-1=a2b-1 >
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C4xD4, C8.26D4
Character table of C8.26D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -i | -1 | 1 | i | -i | i | i | -i | -1 | -i | -i | 1 | i | i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -1 | -1 | -i | -i | i | i | -i | 1 | i | -i | -1 | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | 1 | -1 | i | -i | i | i | -i | -1 | i | i | 1 | -i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | 1 | 1 | -i | -i | i | i | -i | 1 | -i | i | -1 | -i | i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | 1 | 1 | i | i | -i | -i | i | 1 | i | -i | -1 | i | -i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | 1 | -1 | -i | i | -i | -i | i | -1 | -i | -i | 1 | i | i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -1 | -1 | i | i | -i | -i | i | 1 | -i | i | -1 | -i | i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | i | -1 | 1 | -i | i | -i | -i | i | -1 | i | i | 1 | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T113
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)
(1 15 3 9 5 11 7 13)(2 12 4 14 6 16 8 10)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,15,3,9,5,11,7,13)(2,12,4,14,6,16,8,10)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,15,3,9,5,11,7,13)(2,12,4,14,6,16,8,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12)], [(1,15,3,9,5,11,7,13),(2,12,4,14,6,16,8,10)]])
G:=TransitiveGroup(16,113);
C8.26D4 is a maximal subgroup of
C42.283C23 D8:11D4 D8.13D4 C8.5S4
D8p:C4: D16:C4 D24:4C4 D24:10C4 D40:10C4 D40:16C4 Q16:F5 D56:4C4 D56:10C4 ...
M4(2).D2p: M4(2).51D4 M4(2)oD8 M4(2).22D6 C24.54D4 M4(2).22D10 C40.50D4 M4(2).22D14 C56.50D4 ...
C8p.(C2xC4): Q32:C4 D8:4Dic3 D8:4Dic5 D8:F5 SD16:2F5 D8:4Dic7 ...
C8.26D4 is a maximal quotient of
SD16:C8 Q16:5C8 D8:5C8 C8:9D8 C8:12SD16 C8:15SD16 C8:9Q16 C8:M4(2) C8.M4(2) C8:3M4(2) D4.3C42 C8.5C42 M4(2).3Q8 C42.28Q8 SD16:2F5 Q16:F5
M4(2).D2p: M4(2).42D4 M4(2).43D4 M4(2).24D4 M4(2).22D6 D24:10C4 C24.54D4 M4(2).22D10 D40:16C4 ...
(CpxD8):C4: C42.116D4 D8:4Dic3 D8:4Dic5 D8:F5 D8:4Dic7 ...
C42.D2p: C42.107D4 D24:4C4 D40:10C4 D56:4C4 ...
Matrix representation of C8.26D4 ►in GL4(F5) generated by
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 2 | 0 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,3,2,0,0,0,0,1,0,0],[2,0,0,0,0,1,0,0,0,0,3,0,0,0,0,4],[0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,0] >;
C8.26D4 in GAP, Magma, Sage, TeX
C_8._{26}D_4 % in TeX
G:=Group("C8.26D4"); // GroupNames label
G:=SmallGroup(64,125);
// by ID
G=gap.SmallGroup(64,125);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=a^2,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^2*b^-1>;
// generators/relations
Export
Subgroup lattice of C8.26D4 in TeX
Character table of C8.26D4 in TeX