Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C72 ⋊ D8 |
| Sylow | C72 ⋊ D24 |
| Sylow | C3 ⋊ #11716033 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11924248 |
| commutator, Fitting, cosocle | C3×C4×C72 | C22 |
| Frattini, hypercenter | C4 | D7×D21 |
| socle | C2×C3×C72 | D4 |
| hypocenter | C3×C72 | D8 |
| LCS | G ▷ C3×C4×C72 ▷ C2×C3×C72 ▷ C3×C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
136 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 32 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:
2^4,3,7^2|a^2=c,c^2=d,b^a=bc,c^b=cd,e^b=e^2,f^a=f^6,f^b=f^6,g^a=f^2g,g^b=g^6222 conjugacy classes — view the full character table.
| maximal subgroup of | #72269315, #72269325, D3×#11716033, … |
| maximal normal subgroup of | #72269315, #72269325, D3×#11716033, … |
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, 2, 3, 7, 7]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);