Finite Group DatabasebetaFinite Group Databasebeta| order | 2646 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32×C7 |
No aliases for this group.
| Fitting | C32×C72 ⋊ C2×C3 |
| Hall | C32⋊#3862 ⋊ C2 |
| Hall | C7⋊D7 ⋊ H(3,𝔽3) |
| Sylow | C72 ⋊ C2×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×#64425 |
| commutator, socle, cosocle | C3×C72 | C2×C32 |
| Fitting | C32×C72 | C2×C3 |
| hypercenter | C32 | #64425 |
| hypocenter | C72 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×C72 ▷ C72 | |
| UCS | C1 ◁ C3 ◁ C32 | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,5,6)(4,8,9)(10,11)(12,13)(14,15)(17,20)(18,22)(19,21);(1,2,4)(3,6,8)(5,9,7)(11,12,16)(13,15,14)(18,19,21)(20,23,22)Polycyclic presentation:
2,3^3,7^2|c^b=cd,e^a=e^6,e^b=f,f^a=f^6,f^b=e^6f^694 conjugacy classes — view the full character table.
| maximal subgroup of | #72224660, #72224674, #72230093, … |
| maximal normal subgroup of | #72224660, #72224674, #72230093, … |
| maximal quotient of | C2×#72224658 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 7, 7]);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.5^6*F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);