Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of #1088 |
| automorphism | Aut(G) = AΓL(1,𝔽9) (order 144) · Inn(G) = #157 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72228630 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C3 | #157 |
| commutator, Fitting, hypocenter | H(3,𝔽3) | C4 |
| prime radical, cosocle | #272 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
17 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,7,9,14)(4,11)(5,8)(6,12,13,18)(10,15,16,17);(1,3,16)(2,15,6)(4,12,9)(5,10,18)(7,17,11)(8,13,14)Polycyclic presentation:
2^2,3^3|a^2=b,c^a=cd^2,d^a=c^2d^2,c^b=c^2,d^b=d^2e,d^c=de14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 15);