Finite Group DatabasebetaFinite Group Databasebeta| order | 4410 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C72 · Schur multiplier (H2) = C3×C7 |
| direct product | C5×C72×C3⋊D3 |
| unambiguous semidirect product | C3⋊(C5×D3×C72) |
| Fitting, socle, Hall | C32×C5×C72 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C5×C72 |
| Hall | C32×C72 ⋊ C2×C5 |
| Hall | C32×C5 ⋊ C2×C72 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C72 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C2×C5×C72 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C72 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C5×C72 | |
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:
(1,2,3,4,5)(6,7,8,9,10,11,12)(21,22)(24,25);(13,14,15,16,17,18,19)(23,24,25);(20,21,22)1470 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C72×C3⋊D3×C25, C52×C72×C3⋊D3, C5×C72×C3⋊D15 |
| maximal normal subgroup of | C72×C3⋊D3×C25, C52×C72×C3⋊D3 |
| maximal quotient of | C72×C3⋊D3×C25, C52×C72×C3⋊D3, C5×C72×C3⋊D15 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4410, 131);