Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C22×C72 ⋊ C22×C32 |
| Hall | #11716115 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ C2×D4 |
| Sylow | C72 ⋊ C2×C32×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽7)2 |
| commutator, socle, cosocle | C2×C72 | C23×C32 |
| Fitting | C22×C72 | C22×C32 |
| hypercenter | C22 | AGL(1,𝔽7)2 |
| hypocenter | C72 | C2×C32×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C22 | |
680 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^3,3^2,2,7^2|a^2=f,b^a=bf,g^a=g^6,g^c=g^6,g^d=g^2,g^e=g^4,h^a=g^6h,h^c=h^6,h^d=g^4h,h^e=h^4124 conjugacy classes — view the full character table.
| maximal subgroup of | #72245383 |
| maximal normal subgroup of | #72245383 |
| maximal quotient of | #72245397, #72245407, #72245450 |
| holomorph of | ΓL(1,𝔽8)⋊C4, C2×AGL(1,𝔽7) |
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, 3, 3, 2, 7, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.7^4*F.8);;
gap> SetConjugate(coll, 8, 5, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);