Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ C3×#5933 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72214032 |
| commutator, cosocle, hypocenter | C7×#997 | C2×C3 |
| Fitting | C7×H(5,𝔽2) | C3×D3 |
| socle | C2×C7 | C3×#1020 |
| prime radical | ΓL(1,𝔽8)×#997 | C2 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 | |
286 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,12,11,7,5,15,3,16)(2,13,9,14,8,10,6,4)(18,21,22,19,20,23);(3,4,13)(6,7,15)(9,16,12)(10,11,14)(17,18,20,23,22,21,19)Polycyclic presentation:
2,3^2,7,2^5|a^2=i,e^2=i,f^2=i,c^a=c^2,d^a=d^6,d^b=d^2,e^a=ei,e^c=f,f^a=ef,f^c=efi,f^e=fi,g^a=h,g^c=h,g^e=gi,h^a=g,h^c=gh,h^e=hi,h^f=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72259503, #72259795, #72259821, … |
| maximal normal subgroup of | #72259503, #72259795, #72259821, … |
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, 3, 3, 7, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);