Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×D4 (order 16) · Inn(G) = C1 · Out(G) = C2×D4 |
| holomorph | G ⋊ Aut(G) = D3×B(4,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C3×C4 |
| multiplicative | (ℤ/35)ˣ, (ℤ/39)ˣ, (ℤ/45)ˣ, (ℤ/52)ˣ, (ℤ/70)ˣ, (ℤ/78)ˣ, (ℤ/90)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C3 |
| socle, prime radical | C22×C3 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,3,4)(7,8,9);(5,6)24 conjugacy classes — view the full character table.
| center of | C3×#97, C3×C22⋊C8, C3×C4⋊C8, … |
| derived subgroup of | #4451, #4452, #4464, … |
| Frattini subgroup of | C4×C9⋊C8, #63362, C4⋊(C9⋊C8), … |
| Fitting subgroup of | C2×C3⋊C8, #208, C4×Dic3, … |
| maximal subgroup of | C2×C3⋊C8, #208, C4×Dic3, … |
| maximal normal subgroup of | C2×C3⋊C8, #208, C4×Dic3, … |
| maximal quotient of | C3×C42, C3×C22⋊C4, C3×C4⋊C4, … |
| automorphism group of | C5×C7, C3×C13, C5×C9, … |
| outer automorphism group of | C5×C7, C3×C13, C5×C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 9);