Finite Group DatabasebetaFinite Group Databasebeta| order | 16 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = B(4,𝔽2) (order 64) · Inn(G) = C22 · Out(G) = C2×D4 |
| holomorph | G ⋊ Aut(G) = #72210342 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| Borel | B(2,ℤ/4) |
| direct product | C2×D4 |
| unambiguous semidirect product | C4⋊C22 |
| unambiguous semidirect product | C22⋊C22 |
| unambiguous semidirect product | C23⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22 |
| commutator, Frattini, cosocle | C2 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 6 points:
(1,2);(4,6);(3,4)(5,6)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 |
| order | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 |
| χ5 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 |
| χ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 |
| χ7 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ8 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 |
| χ9 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 |
| χ10 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 |
| derived subgroup of | #1402, #1403, #2196, … |
| Frattini subgroup of | AΣL(1,𝔽16), #352, #353, … |
| Fitting subgroup of | D4×SL(2,𝔽5), C2×D4×A5, C2×#91881, … |
| maximal subgroup of | #99, #100, #102, … |
| maximal normal subgroup of | #99, #100, #102, … |
| maximal quotient of | C2×C22⋊C4, C2×C4⋊C4, C4×D4, … |
| automorphism group of | C2×C8, M16, SD16, … |
| outer automorphism group of | C2×C8, C2×D4, C2×C3×C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16, 11);