Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C7 |
| unambiguous semidirect product | (C3⋊AGL(1,𝔽11))⋊(C7⋊D7) |
| unambiguous semidirect product | (D33⋊(C7⋊D7))⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C72×C11 | C22×C5 |
| LCS | G ▷ C3×C72×C11 | |
| UCS | C1 | |
400 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1617 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,7,3,9)(4,11,10,5,8)(13,14)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27);(2,3)(4,6)(5,8)(7,11)(9,10)(12,13,14)(16,17)(18,19)(20,21)(23,24)(25,26)(27,28);(15,16,18,20,21,19,17)Polycyclic presentation:
2^2,5,3,7^2,11|e^a=e^6,f^a=f^6,g^a=g^10,d^b=d^2,e^b=e^6,f^b=f^6,g^c=g^4465 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 311);