Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | ΓL(1,𝔽8)2 ⋊ D4 |
| Hall | #87640 ⋊ C32 |
| Sylow | C72 ⋊ C32×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽7)2 |
| commutator, Fitting, socle, cosocle | C2×C72 | C22×C32 |
| hypocenter | C72 | C32×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 | |
198 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^2,3^2,2,7^2|b^2=e,b^a=be,f^a=f^6,f^b=f^6,f^c=f^4,f^d=f^2,g^a=f^6g,g^b=g^6,g^c=f^5g,g^d=g^271 conjugacy classes — view the full character table.
| maximal subgroup of | #72245264, #72245296, #72245303 |
| maximal normal subgroup of | #72245264, #72245296, #72245303 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 2, 7, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.6^5*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);