Finite Group DatabasebetaFinite Group Databasebeta| order | 4 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = D3 (order 6) · Inn(G) = C1 · Out(G) = D3 |
| holomorph | G ⋊ Aut(G) = S4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22 |
| multiplicative | (ℤ/8)ˣ, (ℤ/12)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
5 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4 |
| transitive degree | μt(G) = 4 |
A minimal faithful permutation representation — 2 generators on 4 points:
(1,2);(3,4)| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 |
| order | 1 | 2 | 2 | 2 |
| χ1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | -1 | 1 | -1 |
| χ3 | 1 | 1 | -1 | -1 |
| χ4 | 1 | -1 | -1 | 1 |
| center of | C22⋊C4, C4⋊C4, C2×D4, … |
| derived subgroup of | A4, C2×A4, #99, … |
| Frattini subgroup of | C42, C22⋊C4, C4⋊C4, … |
| Fitting subgroup of | A4, S4, C2×SL(2,𝔽5), … |
| socle of | C2×C4, A4, C42, … |
| maximal subgroup of | C2×C4, D4, C23, … |
| maximal normal subgroup of | C2×C4, D4, C23, … |
| maximal quotient of | C2×C4, D4, Q8, … |
| automorphism group of | C8, C3×C4 |
| outer automorphism group of | C8, C3×C4, M16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4, 2);