Finite Group DatabasebetaFinite Group Databasebeta| order | 16 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = #5934 (order 192) · Inn(G) = C1 · Out(G) = #5934 |
| holomorph | G ⋊ Aut(G) = #72209832 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C22×C4 |
| multiplicative | (ℤ/40)ˣ, (ℤ/48)ˣ, (ℤ/60)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C23 |
| socle, prime radical | C23 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 8 points:
(1,2,3,4);(5,6);(7,8)16 conjugacy classes — view the full character table.
| center of | #336, #376, C4×C22⋊C4, … |
| derived subgroup of | #1343, #1344, #1345, … |
| Frattini subgroup of | #323, #324, #327, … |
| Fitting subgroup of | C4×A4, A4⋊C8, A4⋊Q8, … |
| socle of | #72258185, C4×#62832, C2×#10586400, … |
| maximal subgroup of | #95, C22⋊C8, C2×C42, … |
| maximal normal subgroup of | #95, C22⋊C8, C2×C42, … |
| maximal quotient of | C2×C42, C2×C22⋊C4, C2×C4⋊C4, … |
| automorphism group of | C5×C8, C3×C16, C3×C4×C5 |
| outer automorphism group of | C5×C8, C3×C16, C3×C4×C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16, 10);