Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of #1113 |
| automorphism | Aut(G) = #10620184⋊C2 (order 1296) · Inn(G) = G · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C33 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C33 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2)(5,6)(7,8,9);(2,3)(4,5,6)(8,9)Polycyclic presentation:
2^2,3^3|c^a=c^2,e^a=e^2,c^b=c^2,d^b=d^215 conjugacy classes — view the full character table.
| derived subgroup of | #1113⋊C3, D33⋊C3, C2×#1113⋊C3, … |
| maximal subgroup of | D32⋊D3, #157⋊D3, D33, … |
| maximal normal subgroup of | D32⋊D3, #157⋊D3, D33, … |
| maximal quotient of | #6294, #6295, #6296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 40);