Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C72×D4 ⋊ C2×C3 |
| Hall | C7×#1159 ⋊ C3 |
| Hall | #3861 ⋊ #41 |
| Sylow | C72 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#64423 |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C72×D4 | C2×C3 |
| hypercenter | D4 | #64423 |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ D4 | |
208 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^3,3,2,7^2|b^a=be,c^a=ce,c^b=ce,f^a=f^6,f^d=f^4,g^a=fg,g^d=g^485 conjugacy classes — view the full character table.
| maximal subgroup of | #72214216, #72214244, #72214252, … |
| maximal normal subgroup of | #72214216, #72214244, #72214252, … |
| maximal quotient of | C2×#72214027 |
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, 2, 3, 2, 7, 7]);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);