Finite Group DatabasebetaFinite Group Databasebeta| order | 4410 |
| isoclinism | isoclinic class of C5⋊(C7⋊D7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C7 |
| direct product | C32×C5⋊(C7⋊D7) |
| unambiguous semidirect product | C5⋊(C32×C7⋊D7) |
| Fitting, socle, Hall | C32×C5×C72 ⋊ C2 |
| derived, Hall | C5×C72 ⋊ C2×C32 |
| Hall | C32×C72 ⋊ D5 |
| Sylow | C72 ⋊ C32×D5 |
| Hall | C32×C5 ⋊ C7⋊D7 |
| Sylow | C5 ⋊ C32×C7⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C5⋊(C7⋊D7) |
| commutator, cosocle, hypocenter | C5×C72 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C72 | C2 |
| LCS | G ▷ C5×C72 | |
| UCS | C1 ◁ C32 | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 2205 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(8,9)(10,11)(13,14)(15,16)(17,18)(20,21)(22,23)(24,25);(4,5,6)(7,8,10,11,9)(19,20,22,24,25,23,21);(12,13,15,17,18,16,14)1116 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×C7⋊D175, C32×C5×C5⋊(C7⋊D7), C32×C5⋊(C5⋊(C7⋊D7)) |
| maximal normal subgroup of | C32×C5×C5⋊(C7⋊D7) |
| maximal quotient of | C32×C7⋊D175, C32×C5×C5⋊(C7⋊D7), C32×C5⋊(C5⋊(C7⋊D7)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4410, 124);