Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived, Sylow | C72 ⋊ C2×C32×C8 |
| Hall | #11716029 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | AGL(1,𝔽7)2 |
| commutator, hypocenter | C72 | C2×C32×C8 |
| Fitting, cosocle | C4×C72 | C22×C32 |
| socle | C2×C72 | C2×C32×C4 |
| prime radical | C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7) | C4 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C4 | |
280 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 85 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:
2^2,3^2,2^2,7^2|a^2=e,e^2=f,g^a=g^6,g^b=g^6,g^c=g^4,g^d=g^2,h^a=g^6h,h^b=h^6,h^c=g^5h,h^d=h^2196 conjugacy classes — view the full character table.
| maximal subgroup of | #72245201, #72245397, #72245407 |
| maximal normal subgroup of | #72245201, #72245397, #72245407 |
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, 3, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.7^5*F.8);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);