Finite Group DatabasebetaFinite Group Databasebeta| order | 16 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = GL(2,ℤ/4) (order 96) · Inn(G) = C1 · Out(G) = GL(2,ℤ/4) |
| holomorph | G ⋊ Aut(G) = AGL(2,ℤ/4) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C4 |
| abelian | C42 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, socle, prime radical, cosocle | C22 | C22 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
15 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2,3,4);(5,6,7,8)16 conjugacy classes — view the full character table.
| center of | #322, C4×M16, #431, … |
| derived subgroup of | C42⋊C3, C2×C42⋊C3, #126⋊C3, … |
| Frattini subgroup of | C82, #322, D4⋊C8, … |
| Fitting subgroup of | C42⋊C3, #857, C42×A5, … |
| socle of | #72258025 |
| maximal subgroup of | C4×C8, #97, #104, … |
| maximal normal subgroup of | C4×C8, #97, #104, … |
| maximal quotient of | #95, C4×C8, #97, … |
| outer automorphism group of | C5×D15, #3932, #3958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16, 2);