Finite Group DatabasebetaFinite Group Databasebeta| order | 3654 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C29 · Schur multiplier (H2) = C1 |
| direct product | C29×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C9×C29) |
| unambiguous semidirect product | (C7⋊C9)⋊(C2×C29) |
| unambiguous semidirect product | (D7×C29)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C9×C29 |
| Hall | C29×C7⋊C9 ⋊ C2 |
| Hall | D7×C29 ⋊ C9 |
| Hall | C7×C29 ⋊ C2×C9 |
| Hall | C7⋊C9 ⋊ C2×C29 |
| Hall | D7 ⋊ C9×C29 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C29 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C9×C29 |
| Frattini | C3 | C29×AGL(1,𝔽7) |
| Fitting, socle | C3×C7×C29 | C2×C3 |
| prime radical | C3×D7×C29 | C3 |
| cosocle | C3×C7 | C2×C3×C29 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C29 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1827 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(31,35,34,32,33,36)(37,41,45,39,44,40,42,38,43);(30,31,34,35,33,36,32)609 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×C29×D7⋊C9, C29×C5⋊(D7⋊C9), C29×D49⋊C9, … |
| maximal normal subgroup of | C5×C29×D7⋊C9, D7⋊C9×C29⋊C7, C7×C29×D7⋊C9 |
| maximal quotient of | C5×C29×D7⋊C9, C29×C5⋊(D7⋊C9), C29×D49⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3654, 7);