Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting, derived, Sylow | C32⋊H(5,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72277662 |
| commutator, Fitting, cosocle, hypocenter | C32⋊H(5,𝔽3) | C22 |
| Frattini, socle | C32 | C3⋊D3×C3⋊(C3⋊D3) |
| LCS | G ▷ C32⋊H(5,𝔽3) | |
| UCS | C1 ◁ C3 | |
8245 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 8245 conjugacy classes of subgroups — too many to draw. Only its 221 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:
(11,14)(12,29,20,18,19,30)(13,21)(15,22)(16,33,28,26,25,36)(17,24)(23,35,31,27,32,34);(2,4)(5,9)(6,8)(10,12,18)(11,16,27)(13,19,29,21,20,30)(14,23,26)(15,28,34,17,25,35)(22,32,33,24,31,36);(2,5,6)(4,8,9)(10,11,24)(12,16,32)(13,17,14)(15,22,21)(18,27,36)(19,25,31)(20,28,23)(26,29,35)(30,34,33);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2^2,3^7|c^b=c^2,d^a=d^2,e^a=e^2,e^c=eh,e^d=ei,f^b=f^2,g^a=g^2,g^f=gh,h^a=h^2,h^b=h^2135 conjugacy classes — view the full character table.
| maximal subgroup of | #72277883 |
| maximal normal subgroup of | #72277883 |
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", "i");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);