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) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×Dic7 |
| derived, Sylow | C7 ⋊ C32×C4 |
| Hall | C32×C7 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D7 |
| commutator, hypocenter | C7 | C32×C4 |
| Frattini | C2 | C32×D7 |
| Fitting, socle, prime radical | C2×C32×C7 | C2 |
| cosocle | C2×C7 | C2×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C32 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3)(7,8)(9,10)(11,12,13,14,15,16,17);(4,5,6)(7,9,8,10)(12,17)(13,16)(14,15)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×Dic7, C3×D21⋊C4, C3×#4247, … |
| maximal normal subgroup of | C3×D3×Dic7, C3×D21⋊C4, C3×#4247, … |
| maximal quotient of | C32×C7⋊C8, C2×C32×Dic7, C3×C9×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 20);