Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | #11716103 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ #41 |
| Sylow | C72 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C4×C72 | C22×C3 |
| hypercenter | C4 | D7×AGL(1,𝔽7) |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
168 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 55 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:
2^3,3,2,7^2|a^2=e,c^2=e,c^b=ce,f^a=f^6,f^b=f^6,f^d=f^4,g^a=f^6g,g^b=g^6,g^d=f^5g119 conjugacy classes — view the full character table.
| maximal subgroup of | #72245780, #72245794, #72245818, … |
| maximal normal subgroup of | #72245780, #72245794, #72245818, … |
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> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.6^5*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);