Finite Group DatabasebetaFinite Group Databasebeta| order | 1188 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×((C3⋊D3)⋊C3)⋊C2 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊(C2×C11) |
| unambiguous semidirect product | #272⋊(C2×C11) |
| Fitting, Hall | C11×H(3,𝔽3) ⋊ C22 |
| derived, Sylow | H(3,𝔽3) ⋊ C22×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C22×C11 |
| Frattini | C3 | D32×C11 |
| Fitting | C11×H(3,𝔽3) | C22 |
| socle | C3×C11 | D32 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C11 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 99 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,19,17,15,16,20)(14,18);(12,13,15)(14,16,19,18,17,20)121 conjugacy classes — view the full character table.
| maximal subgroup of | C11×AGL(2,𝔽3), C11⋊C5×((C3⋊D3)⋊C3)⋊C2, D11×((C3⋊D3)⋊C3)⋊C2 |
| maximal normal subgroup of | C11⋊C5×((C3⋊D3)⋊C3)⋊C2, D11×((C3⋊D3)⋊C3)⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1188, 38);