Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #14552587 ⋊ C2 |
1020 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,5,2,7,9,8,4,3,6)(11,17,18,13,14,15)(12,16)(20,26,27,22,23,24)(21,25);(1,6,9,4,8,5,7,2,3)(19,23,22,21,27,24,25,20,26);(10,14,13,12,18,15,16,11,17)Polycyclic presentation:
2,3^7|b^3=d,c^3=d^2,e^3=g,f^3=h,c^b=cd^2,e^a=e^2g^2,e^c=eg,f^a=f^2h^2,f^c=fh,g^a=g^2,h^a=h^2174 conjugacy classes — view the full character table.
| maximal subgroup of | 3−1+2×#92314, #72243278, #72243293, … |
| maximal normal subgroup of | 3−1+2×#92314, #72243278, #72243293, … |
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, 2, F.4);;
gap> SetPower(coll, 3, F.4^2);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);