Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Sylow | C32⋊H(5,𝔽3) ⋊ C22 |
| derived | H(3,𝔽3)2 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72243929 |
| commutator, cosocle, hypocenter | H(3,𝔽3)2 | C22×C3 |
| Frattini, socle | C32 | C3×C3⋊#1113 |
| Fitting | C32⋊H(5,𝔽3) | C22 |
| LCS | G ▷ H(3,𝔽3)2 | |
| UCS | C1 ◁ C3 | |
4843 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 4843 conjugacy classes of subgroups — too many to draw. Only its 95 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 — 3 generators on 36 points:
(2,8)(4,5)(6,9)(10,16,30,15,19,36)(11,18,34,13,28,17)(12,27,20,25,29,24)(14,32,26,23,22,35)(21,31,33);(2,4)(5,9)(6,8)(11,14)(12,29,19,17,18,30)(13,20)(15,21)(16,35,25,26,28,33)(22,34,32,27,31,36)(23,24);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2^2,3^7|d^a=d^2,d^b=d^2,e^a=e^2,e^b=e^2,e^c=eh,e^d=ei,f^a=f^2,g^b=g^2,g^f=gh,h^a=h^2,h^b=h^2171 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, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
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.9);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 2, 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);