Finite Group DatabasebetaFinite Group Databasebeta| order | 41352 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D20676 |
| Fitting | C3×C4×C1723 ⋊ C2 |
| Hall | C3×C1723 ⋊ D4 |
| Sylow | C1723 ⋊ D12 |
| Sylow | C3 ⋊ D6892 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5169 |
| commutator, socle, cosocle | C2×C3×C1723 | C22 |
| Fitting | C3×C4×C1723 | C2 |
| hypercenter | C4 | D5169 |
| hypocenter | C3×C1723 | D4 |
| LCS | G ▷ C2×C3×C1723 ▷ C3×C1723 | |
| UCS | C1 ◁ C2 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1730 |
| transitive degree | μt(G) = 20676 |
A minimal faithful permutation representation — 2 generators on 1730 points:
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presentation:
2^3,3,1723|b^2=c,b^a=bc,d^a=d^2,e^a=e^172210341 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 1723]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^1722);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);