p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.C8, C8.24D4, M5(2)⋊3C2, C4.7M4(2), (C2×C4).C8, (C2×C8).3C4, (C22×C4).4C4, C22.4(C2×C8), C2.7(C22⋊C8), (C2×C8).42C22, C4.29(C22⋊C4), (C2×M4(2)).10C2, (C2×C4).67(C2×C4), SmallGroup(64,30)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.C8
G = < a,b,c,d | a2=b2=c2=1, d8=c, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >
Character table of C23.C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 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 | -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 | 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 | -i | i | -i | -i | i | i | i | -i | linear of order 4 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | ζ8 | ζ83 | ζ8 | ζ85 | ζ83 | ζ87 | ζ87 | ζ85 | linear of order 8 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -i | i | i | -i | ζ85 | ζ83 | ζ8 | ζ85 | ζ87 | ζ87 | ζ83 | ζ8 | linear of order 8 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | ζ83 | ζ8 | ζ83 | ζ87 | ζ8 | ζ85 | ζ85 | ζ87 | linear of order 8 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | ζ87 | ζ85 | ζ87 | ζ83 | ζ85 | ζ8 | ζ8 | ζ83 | linear of order 8 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | i | -i | -i | i | ζ83 | ζ85 | ζ87 | ζ83 | ζ8 | ζ8 | ζ85 | ζ87 | linear of order 8 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -i | i | i | -i | ζ8 | ζ87 | ζ85 | ζ8 | ζ83 | ζ83 | ζ87 | ζ85 | linear of order 8 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | ζ85 | ζ87 | ζ85 | ζ8 | ζ87 | ζ83 | ζ83 | ζ8 | linear of order 8 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | i | -i | -i | i | ζ87 | ζ8 | ζ83 | ζ87 | ζ85 | ζ85 | ζ8 | ζ83 | linear of order 8 |
| ρ17 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 0 | -2 | -2 | 2 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | 2 | -2 | 0 | -2 | -2 | 2 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T104
(2 10)(3 11)(6 14)(7 15)
(2 10)(4 12)(6 14)(8 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (2,10)(3,11)(6,14)(7,15), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (2,10)(3,11)(6,14)(7,15), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(2,10),(3,11),(6,14),(7,15)], [(2,10),(4,12),(6,14),(8,16)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,104);
C23.C8 is a maximal subgroup of
C24.C8 C23.1M4(2) C42.C8 C22⋊C4.C8 C23.D8 C23.2D8 C23.SD16 C23.2SD16 M5(2).19C22 D8⋊D4 D8.D4 M5(2).C22 C23.10SD16
C4p.M4(2): C8.5M4(2) C8.19M4(2) C8.25D12 C24.D4 C8.25D20 C40.D4 C20.10M4(2) C20.29M4(2) ...
C23.C8 is a maximal quotient of
C23⋊C16 C22.M5(2) C8.17Q16 C20.10M4(2)
C8.D4p: C8.31D8 C8.25D12 C8.25D20 M5(2)⋊D7 ...
(C2×C4p).C8: M5(2)⋊C4 C24.D4 C40.D4 C20.29M4(2) C56.D4 ...
Matrix representation of C23.C8 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,1,3,0,0,0] >;
C23.C8 in GAP, Magma, Sage, TeX
C_2^3.C_8
% in TeX
G:=Group("C2^3.C8"); // GroupNames label
G:=SmallGroup(64,30);
// by ID
G=gap.SmallGroup(64,30);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,650,489,69,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=1,d^8=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export
Subgroup lattice of C23.C8 in TeX
Character table of C23.C8 in TeX