Finite Group DatabasebetaFinite Group Databasebeta| order | 3136 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C7 |
No aliases for this group.
| Fitting | C72×#99 ⋊ C2 |
| Sylow | C72 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #13096523 |
| commutator, cosocle | C23×C72 | C23 |
| Frattini | C23 | C22×C7⋊D7 |
| Fitting | C72×#99 | C2 |
| socle | C2×C72 | #120 |
| hypercenter | #99 | C7⋊D7 |
| hypocenter | C72 | B(4,𝔽2) |
| LCS | G ▷ C23×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #99 | |
990 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 990 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 392 |
Polycyclic presentation:
2^6,7^2|a^2=f,c^a=ce,c^b=cd,d^a=df,e^b=ef,g^b=g^6,g^c=g^6,h^b=h^6,h^c=h^6280 conjugacy classes — view the full character table.
| maximal subgroup of | B(4,𝔽2)×C7⋊D7, #72246176, #72246191, … |
| maximal normal subgroup of | B(4,𝔽2)×C7⋊D7, #72246176, #72246191, … |
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, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);