Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C2×C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | #11744560 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ H(7,𝔽2) |
| Sylow | C7 ⋊ C3×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C26×C3 |
| Fitting | C2×C7×H(5,𝔽2) | C2×C3 |
| hypercenter | C2×H(5,𝔽2) | AGL(1,𝔽7) |
| hypocenter | C7 | C3×H(7,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×H(5,𝔽2) | |
11984 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^6,3,2,7|c^2=h,d^2=h,b^a=bh,d^c=dh,e^d=eh,f^c=fh,f^d=fh,i^b=i^6,i^g=i^2229 conjugacy classes — view the full character table.
| maximal subgroup of | #72259707 |
| maximal normal subgroup of | #72259707 |
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, 2, 2, 2, 2, 3, 2, 7]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);