Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C35 |
No aliases for this group.
| Fitting, Sylow | #14555282 ⋊ C2 |
| derived | #10624862 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C3⋊(C3⋊AGL(1,ℤ/9))) |
| commutator, cosocle, hypocenter | #10624862 | C2×C3 |
| Frattini, socle | C32 | C3×C3⋊(C3⋊(C3⋊D3)) |
| Fitting | #14555282 | C2 |
| LCS | G ▷ #10624862 | |
| UCS | C1 ◁ C3 | |
4852 conjugacy classes of subgroups; only the 615 normal subgroups are shown.
This group has 4852 conjugacy classes of subgroups — too many to draw. Only its 615 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 — 5 generators on 36 points:
(2,8,9,4,5,6)(3,7)(11,14)(12,18)(15,24)(16,26)(17,22)(19,30)(20,29)(23,28)(25,33)(27,35)(31,36)(32,34);(1,5,4,3,9,6,7,2,8)(10,12,18)(11,16,28)(13,20,29)(14,23,26)(15,25,34)(17,27,36)(19,30,21)(22,31,35)(24,32,33);(12,20,19)(16,27,25)(18,30,29)(23,31,32)(26,33,35)(28,34,36);(11,15,17)(12,19,20)(14,22,24)(16,27,25)(18,29,30)(26,33,35);(10,11,14)(12,16,23)(13,17,22)(15,24,21)(18,28,26)(19,25,32)(20,27,31)(29,36,35)(30,34,33)Polycyclic presentation:
2,3^7|f^3=g,c^a=c^2,d^a=d^2,d^c=dh,e^a=e^2,f^a=f^2g^2,f^b=fg,f^d=fh^2,f^e=fh,g^a=g^2171 conjugacy classes — view the full character table.
| maximal subgroup of | #72253822 |
| maximal normal subgroup of | #72253822 |
| maximal quotient of | #72243459 |
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, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);