Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽3)2 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11758091 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽3)2 | C22 |
| Frattini, socle | C32 | C3⋊#1113 |
| LCS | G ▷ H(3,𝔽3)2 | |
| UCS | C1 ◁ C3 | |
1302 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,6,5)(4,9,8)(10,11,13)(12,15,16,18,14,17);(2,4)(5,9)(6,8)(10,12,15)(11,16,17,13,14,18);(1,2,4)(3,6,8)(5,9,7)(10,13,11)(12,16,14)(15,18,17)Polycyclic presentation:
2^2,3^6|c^a=c^2,c^b=c^2h^2,d^a=d^2,d^b=cdh^2,e^b=e^2,e^c=eh,e^d=eh,f^a=e^2f^2,f^b=ef,f^c=fh,f^d=fg,g^a=gh,h^a=h^273 conjugacy classes — view the full character table.
| maximal subgroup of | #272×((C3⋊D3)⋊C3)⋊C2, #72228697, #72259777, … |
| maximal normal subgroup of | #272×((C3⋊D3)⋊C3)⋊C2, #72228697, #72259777, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3^2*F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);