Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = D3×#272 (order 324) · Inn(G) = C32 · Out(G) = D32 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2 |
| direct product | C22×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32 |
| commutator, Frattini, cosocle | C3 | C22×C32 |
| prime radical | C22×C32 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(5,6,8,7,10,12,11,9,13);(3,4)(6,9,10)(8,12,13)44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 31);