Finite Group DatabasebetaFinite Group Databasebeta| order | 3969 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting | C32×C72 ⋊ C32 |
| Sylow | C72 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×ΓL(1,𝔽8)2 |
| commutator, socle, cosocle | C3×C72 | C33 |
| Fitting | C32×C72 | C32 |
| hypercenter | C32 | ΓL(1,𝔽8)2 |
| hypocenter | C72 | C3×H(3,𝔽3) |
| LCS | G ▷ C3×C72 ▷ C72 | |
| UCS | C1 ◁ C3 ◁ C32 | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,5,6)(4,8,9)(10,14,16,12,15,11,13)(17,18,19,20,21,23,22);(11,12,13)(14,16,15)(18,19,21)(20,22,23);(1,2,4)(3,6,8)(5,9,7)(18,19,21)(20,22,23)Polycyclic presentation:
3^4,7^2|c^a=cd,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2113 conjugacy classes — view the full character table.
| maximal subgroup of | #72224660, #72224666, #72224671, … |
| maximal normal subgroup of | #72224660, #72224666, #72224671, … |
| maximal quotient of | C2×#72224636 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 7, 7]);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);