Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32×C9 |
No aliases for this group.
| Fitting, Sylow | #14552609 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×#92309 |
| commutator, cosocle | C3×C92 | C2×C32 |
| Frattini, socle | C33 | C32×C3⋊D3 |
| Fitting | #14552609 | C2 |
| hypercenter | C32 | #92309 |
| hypocenter | C92 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×C92 ▷ C92 | |
| UCS | C1 ◁ C3 ◁ C32 | |
916 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 916 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 486 |
Polycyclic presentation:
2,3^7|e^3=g,f^3=h,c^b=cd,e^a=e^2g^2,e^b=eg^2,f^a=f^2h^2,f^b=fg^2h,g^a=g^2,h^a=h^2174 conjugacy classes — view the full character table.
| maximal subgroup of | #72243669, #72243965, #72244044, … |
| maximal normal subgroup of | #72243669, #72243965, #72244044, … |
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, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7^2*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);