Finite Group DatabasebetaFinite Group Databasebeta| order | 4 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2 (order 2) · Inn(G) = C1 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = D4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C4 |
| multiplicative | (ℤ/5)ˣ, (ℤ/10)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, socle, prime radical, cosocle | C2 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
3 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4 |
| transitive degree | μt(G) = 4 |
A minimal faithful permutation representation — 1 generator on 4 points:
(1,2,3,4)| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 |
| order | 1 | 4 | 2 | 4 |
| χ1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | -1 | ||
| χ3 | 1 | -1 | 1 | -1 |
| χ4 | 1 | -1 |
Here i is the imaginary unit (i² = −1).
| center of | M16, #41, C3⋊C8, … |
| derived subgroup of | D8, SD16, Q16, … |
| Frattini subgroup of | C8, C2×C8, M16, … |
| Fitting subgroup of | C4×A5, #6732, A5⋊C8, … |
| socle of | #72258201, #72274291, #72290098 |
| maximal subgroup of | C8, C2×C4, D4, … |
| maximal normal subgroup of | C8, C2×C4, D4, … |
| maximal quotient of | C8, C2×C4, Dic3, … |
| automorphism group of | C5, C2×C5 |
| outer automorphism group of | C5, C2×C5, C5×D3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4, 1);