Finite Group DatabasebetaFinite Group Databasebeta| order | 3024 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting | C3×C7×Q8 ⋊ C3×D3 |
| derived | C3×C7×SL(2,𝔽3) ⋊ C2×C3 |
| Sylow | C7 ⋊ C3×C3⋊GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(A4⋊AGL(1,𝔽7)) |
| commutator, cosocle, hypocenter | C3×C7×SL(2,𝔽3) | C2×C3 |
| Fitting | C3×C7×Q8 | C3×D3 |
| socle | C2×C3×C7 | C3×S4 |
| LCS | G ▷ C3×C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
234 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,7,6,3,4,2,8,5)(10,13,14,11,12,15)(17,18);(2,6,3)(5,7,8)(9,10,12,15,14,13,11);(2,6,3)(5,7,8)(16,17,18)Polycyclic presentation:
2,3^3,7,2^3|f^2=h,g^2=h,c^a=c^2,d^a=d^2,e^a=e^6,e^b=e^2,f^a=gh,f^c=fg,f^d=fg,g^a=fh,g^c=f,g^d=f,g^f=gh66 conjugacy classes — view the full character table.
| maximal subgroup of | #72246260, AGL(1,𝔽7)×C3⋊GL(2,𝔽3), #72269469 |
| maximal normal subgroup of | AGL(1,𝔽7)×C3⋊GL(2,𝔽3) |
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, 3, 3, 3, 7, 2, 2, 2]);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 4, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);