Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C7×#379 ⋊ C2×C3 |
| Hall | #11731297 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #2403 |
| Sylow | C7 ⋊ C3×#2403 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C23×C7 | C24×C3 |
| Fitting | C7×#379 | C2×C3 |
| hypercenter | #379 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#2403 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #379 | |
3128 conjugacy classes of subgroups; only the 337 normal subgroups are shown.
This group has 3128 conjugacy classes of subgroups — too many to draw. Only its 337 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(10,12)(13,14,19)(15,16,18);(6,8);(2,4)(9,10)(11,12)(14,15)(16,18)(17,19);(1,2)(3,4)(5,6)(7,8)(14,15)(16,18)(17,19)Polycyclic presentation:
2^4,3,2^3,7|b^a=bf,c^a=cg,d^b=dh,i^a=i^6,i^b=i^6,i^e=i^4142 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#2403, #72277885, #72277956, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#2403, #72277885, #72277956, … |
| maximal quotient of | #72276708, #72276889 |
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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 7]);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);