Finite Group DatabasebetaFinite Group Databasebeta| order | 8 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = D4 (order 8) · Inn(G) = C22 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = #453 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D4 |
| extraspecial | 2+1+2 |
| affine general linear | AGL(1,ℤ/4) |
| Borel | B(3,𝔽2) |
| unitriangular | UT(3,𝔽2) |
| affine special semilinear | AΣL(1,𝔽4) |
| Heisenberg | H(3,𝔽2) |
| unambiguous semidirect product | C4⋊C2 |
| unambiguous semidirect product | C22⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
8 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);(2,4)Polycyclic presentation:
2^3|b^a=bc| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 |
| order | 1 | 2 | 2 | 2 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | -1 | -1 | 1 |
| χ5 | 2 | -2 | 0 | 0 | 0 |
| Fitting subgroup of | D4×A5, #91881, #11751518, … |
| maximal subgroup of | D8, SD16, C2×D4, … |
| maximal normal subgroup of | D8, SD16, C2×D4, … |
| maximal quotient of | C22⋊C4, C4⋊C4, D8, … |
| automorphism group of | C2×C4, D4 |
| outer automorphism group of | C2×C4, C22⋊C4, C4⋊C4, … |
| holomorph of | C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8, 3);