Finite Group DatabasebetaFinite Group Databasebeta| order | 16 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×S4 (order 48) · Inn(G) = C22 · Out(G) = C2×D3 |
| holomorph | G ⋊ Aut(G) = #11715828 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22 |
| commutator, Frattini, socle, cosocle | C2 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 8 points:
(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^4|c^2=d,b^a=bd| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| order | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 |
| 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 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ||
| χ10 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| derived subgroup of | #576⋊C3, #577⋊C3, #578⋊C3, … |
| Fitting subgroup of | #41⋊C3, GU(2,𝔽3), #984, … |
| maximal subgroup of | #104, #131, #135, … |
| maximal normal subgroup of | #104, #131, #135, … |
| maximal quotient of | #117, C4×D4, C4×Q8, … |
| outer automorphism group of | #2171, D5⋊C16, C2×#674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16, 13);