Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×AGL(1,𝔽7)×GL(2,𝔽3) (order 4032) · Inn(G) = D7 · Out(G) = C2×C3×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C32×D7 |
| Fitting, socle | C2×C32×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C32 |
| Hall | C32×C7 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C32 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(3,4,5)(10,11)(12,13)(14,15);(1,2)(6,7,8)(9,10,12,14,15,13,11)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×D7, C3×#4245, C3×#4246, … |
| maximal normal subgroup of | C3×Dic3×D7, C3×#4245, C3×#4246, … |
| maximal quotient of | C32×Dic14, C32×C4×D7, C32×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 41);