Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×GL(3,𝔽2) (order 336) · Inn(G) = C1 · Out(G) = C2×GL(3,𝔽2) |
| holomorph | G ⋊ Aut(G) = D3×AGL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23×C3 |
| multiplicative | (ℤ/56)ˣ, (ℤ/72)ˣ, (ℤ/84)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 9 points:
(1,2)(7,8,9);(3,4);(5,6)24 conjugacy classes — view the full character table.
| center of | C3×#95, C2×C3×C22⋊C4, C2×C3×C4⋊C4, … |
| derived subgroup of | #4471, #4472, #4475, … |
| Frattini subgroup of | #95⋊C9, #63365, C9⋊#95, … |
| Fitting subgroup of | #217, C22×Dic3, C2×#67, … |
| socle of | C22×Dic3, C22×C3×C4, C23×D3, … |
| maximal subgroup of | #217, C3×C22⋊C4, C22×Dic3, … |
| maximal normal subgroup of | #217, C3×C22⋊C4, C22×Dic3, … |
| maximal quotient of | C22×C3×C4, C2×C3×D4, C2×C3×Q8, … |
| automorphism group of | C7×C8, C8×C9, C3×C4×C7 |
| outer automorphism group of | C7×C8, C8×C9, C3×C4×C7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 15);