Finite Group DatabasebetaFinite Group Databasebeta| order | 16 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×D4 (order 16) · Inn(G) = C1 · Out(G) = C2×D4 |
| holomorph | G ⋊ Aut(G) = #23882 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C8 |
| multiplicative | (ℤ/32)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C22 |
| socle, prime radical | C22 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
11 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,3,4,5,6,7,8);(9,10)16 conjugacy classes — view the full character table.
| center of | #346, C22⋊C16, C4⋊C16, … |
| derived subgroup of | #1339, #1340, #1341, … |
| Frattini subgroup of | C4×C16, #346, #347, … |
| Fitting subgroup of | C8×SL(2,𝔽5), C2×C8×A5, C2×#91145, … |
| maximal subgroup of | C4×C8, #97, C22⋊C8, … |
| maximal normal subgroup of | C4×C8, #97, C22⋊C8, … |
| maximal quotient of | C4×C8, C22⋊C8, C4⋊C8, … |
| automorphism group of | C32 |
| outer automorphism group of | C32, D32, Q64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16, 5);