Finite Group DatabasebetaFinite Group Databasebeta| order | 88 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C5×S4 (order 240) · Inn(G) = C22 · Out(G) = C2×C5×D3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽11)×#5935 |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C1 |
| direct product | Q8×C11 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C11 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C11 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ G | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 88 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,6)(3,7,8,5)(9,10,11,12,13,14,15,16,17,18,19);(1,3,4,8)(2,5,6,7)55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(88, 10);