direct product, p-group, elementary abelian, monomial, rational
Aliases: C24, SmallGroup(16,14)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C24 |
| C1 — C24 |
| C1 — C24 |
Generators and relations for C24
G = < a,b,c,d | a2=b2=c2=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 67, all normal (2 characteristic)
C1, C2, C22, C23, C24
Quotients: C1, C2, C22, C23, C24
Character table of C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ12 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ14 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
►Regular action on 16 points - transitive group 16T3
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 15)(2 16)(3 8)(4 7)(5 12)(6 11)(9 13)(10 14)
(1 5)(2 6)(3 13)(4 14)(7 10)(8 9)(11 16)(12 15)
(1 3)(2 4)(5 13)(6 14)(7 16)(8 15)(9 12)(10 11)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,8)(4,7)(5,12)(6,11)(9,13)(10,14), (1,5)(2,6)(3,13)(4,14)(7,10)(8,9)(11,16)(12,15), (1,3)(2,4)(5,13)(6,14)(7,16)(8,15)(9,12)(10,11)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,8)(4,7)(5,12)(6,11)(9,13)(10,14), (1,5)(2,6)(3,13)(4,14)(7,10)(8,9)(11,16)(12,15), (1,3)(2,4)(5,13)(6,14)(7,16)(8,15)(9,12)(10,11) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,15),(2,16),(3,8),(4,7),(5,12),(6,11),(9,13),(10,14)], [(1,5),(2,6),(3,13),(4,14),(7,10),(8,9),(11,16),(12,15)], [(1,3),(2,4),(5,13),(6,14),(7,16),(8,15),(9,12),(10,11)]])
G:=TransitiveGroup(16,3);
C24 is a maximal subgroup of
C22≀C2 C22⋊A4 C24⋊C5
C24 is a maximal quotient of 2+ 1+4 2- 1+4
Matrix representation of C24 ►in GL4(ℤ) generated by
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,-1] >;
C24 in GAP, Magma, Sage, TeX
C_2^4
% in TeX
G:=Group("C2^4"); // GroupNames label
G:=SmallGroup(16,14);
// by ID
G=gap.SmallGroup(16,14);
# by ID
G:=PCGroup([4,-2,2,2,2]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations
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