Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of C3⋊D21 |
| automorphism | Aut(G) = C2×AGL(1,𝔽7)×AGL(2,𝔽3) (order 36288) · Inn(G) = C3⋊D21 · Out(G) = C2×C3×GL(2,𝔽3) |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊Dic21 |
| unambiguous semidirect product | C7⋊(C3⋊Dic3) |
| derived, Hall | C32×C7 ⋊ C4 |
| Sylow | C32 ⋊ Dic7 |
| Sylow | C7 ⋊ C3⋊Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D21 |
| commutator, hypocenter | C32×C7 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C32×C7 | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,3)(4,5)(6,7)(8,9,10,11)(13,14)(16,17);(1,2,4,6,7,5,3)(15,16,17);(12,13,14)Polycyclic presentation:
2^2,3^2,7|a^2=b,c^a=c^2,d^a=d^2,e^a=e^666 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊Dic3, C3⋊D3×Dic7, #92676, … |
| maximal normal subgroup of | D7×C3⋊Dic3, C3⋊D3×Dic7, #92676, … |
| maximal quotient of | C3⋊(C3⋊(C7⋊C8)), C2×C3⋊Dic21, C3⋊Dic63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 24);