Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, Sylow | #14554289 ⋊ C2 |
| derived | #10624740 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊#92333 |
| commutator, cosocle, hypocenter | #10624740 | C2×C3 |
| Frattini | C33 | C3×C3⋊(C3⋊D3) |
| Fitting | #14554289 | C2 |
| socle | C32 | C3×C3⋊#272 |
| LCS | G ▷ #10624740 | |
| UCS | C1 ◁ C3 | |
2749 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 2749 conjugacy classes of subgroups — too many to draw. Only its 105 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)(11,17,19,15,16,25)(12,20)(13,14)(18,35,21,28,26,33)(22,34)(23,32)(24,36,29,30,27,31);(2,5,6)(4,8,9);(10,12,20)(11,18,30)(13,23,34)(14,22,32)(15,24,28)(16,26,36)(17,27,35)(19,21,31)(25,29,33);(1,2,4)(3,6,8)(5,9,7)(10,19,15,14,16,17,13,11,25)(12,26,29,22,18,24,23,21,27)(20,36,33,32,30,28,34,31,35)Polycyclic presentation:
2,3^7|c^3=h,c^a=c^2h^2,c^b=ch^2,d^a=d^2,d^c=dg,e^a=e^2,e^c=ef,g^a=gh^2,g^d=gh,h^a=h^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #72277465 |
| maximal normal subgroup of | #72277465 |
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, 3, F.8);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.8^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);