Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×D4 (order 16) · Inn(G) = C22 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = D3×#453 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D4 |
| unambiguous semidirect product | C4⋊(C2×C3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 7 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 7 points:
(1,2)(3,4)(5,6,7);(2,4)15 conjugacy classes — view the full character table.
| Fitting subgroup of | #213, #214, D3×D4, … |
| maximal subgroup of | #213, #214, C3×D8, … |
| maximal normal subgroup of | #213, #214, C3×D8, … |
| maximal quotient of | C3×C22⋊C4, C3×C4⋊C4, C3×D8, … |
| outer automorphism group of | C2×Dic7, C2×Dic9, C42⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 10);