Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C72×#41 ⋊ C2×C3 |
| Hall | #13096739 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ H(5,𝔽2) |
| Sylow | C72 ⋊ C3×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×ΓL(1,𝔽8)⋊D7 |
| commutator, socle, cosocle | C2×C72 | C24×C3 |
| Fitting | C72×#41 | C2×C3 |
| hypercenter | #41 | ΓL(1,𝔽8)⋊D7 |
| hypocenter | C72 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ #41 | |
830 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 830 conjugacy classes of subgroups — too many to draw. Only its 204 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:
2^4,3,2,7^2|a^2=f,c^a=cf,d^b=df,g^c=g^6,g^e=h,h^c=h^6,h^e=g^5h^3211 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×ΓL(1,𝔽8)⋊D7, #72246219, #72246291, … |
| maximal normal subgroup of | H(5,𝔽2)×ΓL(1,𝔽8)⋊D7, #72246219, #72246291, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 7, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> SetConjugate(coll, 8, 5, F.7^5*F.8^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);