Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | C32⋊H(5,𝔽3) ⋊ C2 |
| derived | C32×H(3,𝔽3) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊#92391 |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C2×C32 |
| Frattini, socle | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | C32⋊H(5,𝔽3) | C2 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
5851 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 5851 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,9)(6,8)(11,14)(12,27,29,17,18,28)(13,19)(15,20)(16,33,32,25,24,35)(21,36,30,26,31,34)(22,23);(10,12,17)(11,24,36)(13,18,28)(14,30,35)(15,32,26)(16,34,23)(19,29,27)(20,31,25)(21,33,22);(2,5,6)(4,8,9)(10,11,14)(12,16,21)(13,15,22)(17,26,25)(18,24,31)(19,23,20)(27,36,33)(28,34,35)(29,32,30);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2,3^7|d^a=d^2,e^a=e^2,e^b=eg,e^d=eh,f^a=f^2,f^c=fg,g^a=g^2198 conjugacy classes — view the full character table.
| maximal subgroup of | #72277678, #72277681, #72277687, … |
| maximal normal subgroup of | #72277678, #72277681, #72277687, … |
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> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);