Finite Group DatabasebetaFinite Group Databasebeta| order | 3150 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C52×C9 · Schur multiplier (H2) = C5 |
| direct product | C52×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C52×C9) |
| unambiguous semidirect product | (C7⋊C9)⋊(C2×C52) |
| unambiguous semidirect product | (C5×D7)⋊(C5×C9) |
| derived, Sylow | C7 ⋊ C2×C52×C9 |
| Hall | C52×C7⋊C9 ⋊ C2 |
| Hall | C52×D7 ⋊ C9 |
| Hall | C52×C7 ⋊ C2×C9 |
| Hall | C7⋊C9 ⋊ C2×C52 |
| Hall | D7 ⋊ C52×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C52 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C52×C9 |
| Frattini | C3 | C52×AGL(1,𝔽7) |
| Fitting, socle | C3×C52×C7 | C2×C3 |
| prime radical | C3×C52×D7 | C3 |
| cosocle | C3×C7 | C2×C3×C52 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C52 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1575 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5)(12,16,15,13,14,17)(18,22,26,20,25,21,23,19,24);(6,7,8,9,10)(11,12,15,16,14,17,13)525 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C52×D49⋊C9, C52×C7×D7⋊C9, C52×#11725077, … |
| maximal normal subgroup of | C52×C7×D7⋊C9, C52×C11×D7⋊C9 |
| maximal quotient of | C52×D49⋊C9, C52×C7×D7⋊C9, C52×#11725077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3150, 36);