Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3 (order 24) · Inn(G) = D3 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #10595100 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊C8 |
| derived, Sylow | C3 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3 |
| commutator, hypocenter | C3 | C8 |
| Fitting, cosocle | C3×C4 | C2 |
| socle, prime radical | C2×C3 | C4 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(2,3)(4,5,6,8,7,9,10,11);(1,2,3)Polycyclic presentation:
2^3,3|a^2=b,b^2=c,d^a=d^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| order | 1 | 2 | 4 | 4 | 3 | 6 | 12 | 12 | 8 | 8 | 8 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | ||||
| χ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | ||||
| χ5 | 1 | -1 | 1 | -1 | ζ | −ζ | ζ3 | −ζ3 | ||||
| χ6 | 1 | -1 | 1 | -1 | −ζ | ζ | −ζ3 | ζ3 | ||||
| χ7 | 1 | -1 | 1 | -1 | ζ3 | −ζ3 | ζ | −ζ | ||||
| χ8 | 1 | -1 | 1 | -1 | −ζ3 | ζ3 | −ζ | ζ | ||||
| χ9 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| χ10 | 2 | 2 | -2 | -2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 |
| χ11 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | ||||
| χ12 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 8 (ζ8 = 1); i = ζ2.
| maximal subgroup of | D3×C8, ΓL(1,𝔽25), C2×C3⋊C8, … |
| maximal normal subgroup of | D3×C8, ΓL(1,𝔽25), C2×C3⋊C8, … |
| maximal quotient of | C3⋊C16, C2×C3⋊C8, C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 1);