Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, derived, Sylow | C3×#10624749 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×#10624749 | C2 |
| Frattini | C33 | C3⋊(C3⋊(C3⋊D3)) |
| socle | C32 | #92338 |
| LCS | G ▷ C3×#10624749 | |
| UCS | C1 | |
2557 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 2557 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 5 generators on 30 points:
(2,3)(5,9)(6,12)(7,8)(10,19)(11,22)(13,24)(14,16)(15,18)(17,23)(20,26)(21,29)(25,27)(28,30);(4,13,12,8,16,14,7,6,24)(5,21,23,11,25,20,15,10,30)(9,28,19,18,26,27,22,17,29);(6,13,16)(12,14,24)(17,26,28)(20,23,30);(4,5,9)(6,10,17)(7,15,22)(8,11,18)(12,23,19)(13,21,28)(14,20,27)(16,25,26)(24,30,29);(1,2,3)Polycyclic presentation:
2,3^7|e^3=h,b^a=b^2,c^a=c^2,d^a=d^2,d^c=dg,e^a=e^2h^2,e^d=ef,f^a=fh^2,f^c=fh,f^e=fh,g^e=gh^2,h^a=h^299 conjugacy classes — view the full character table.
| maximal subgroup of | #72277778 |
| maximal normal subgroup of | #72277778 |
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, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
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*F.8^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);