Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×D3×#272 (order 648) · Inn(G) = C1 · Out(G) = C2×D3×#272 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C3 |
| abelian | C22×C3×C9 |
| multiplicative | (ℤ/133)ˣ, (ℤ/171)ˣ, (ℤ/189)ˣ, (ℤ/266)ˣ, (ℤ/342)ˣ, (ℤ/378)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C22×C32 |
| socle, prime radical | C22×C32 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(5,6,7,8,9,10,11,12,13);(3,4)(14,15,16)108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 29);