Finite Group DatabasebetaFinite Group Databasebeta| order | 8 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = GL(3,𝔽2) (order 168) · Inn(G) = C1 · Out(G) = GL(3,𝔽2) |
| holomorph | G ⋊ Aut(G) = AGL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23 |
| multiplicative | (ℤ/24)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
16 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);(3,4);(5,6)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 |
| χ3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |
| χ4 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 |
| χ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ6 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 |
| χ7 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ8 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 |
| center of | #95, C2×C22⋊C4, C2×C4⋊C4, … |
| derived subgroup of | AGL(1,𝔽8), AΣL(1,𝔽16), #352, … |
| Frattini subgroup of | #95, #99, C43, … |
| Fitting subgroup of | C2×A4, SL(2,ℤ/4), C2×S4, … |
| socle of | C22×C4, C2×A4, #95, … |
| maximal subgroup of | C22⋊C4, C22×C4, C2×D4, … |
| maximal normal subgroup of | C22⋊C4, C22×C4, C2×D4, … |
| maximal quotient of | C22×C4, C2×D4, C2×Q8, … |
| automorphism group of | C3×C8 |
| outer automorphism group of | C3×C8, #102, #103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8, 5);