Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of C3⋊SO(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊SO(3,𝔽7) |
| unambiguous semidirect product | GL(3,𝔽2)⋊D3 |
| Fitting | C3 ⋊ SO(3,𝔽7) |
| socle, derived | C3×GL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×GL(3,𝔽2) | C2 |
| Fitting | C3 | SO(3,𝔽7) |
| LCS | G ▷ C3×GL(3,𝔽2) | |
| UCS | C1 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,4,6,8,5,2,7,3)(9,10);(1,3,8,6,5,4,7)(9,10,11)15 conjugacy classes — view the full character table.
| maximal subgroup of | D3×SO(3,𝔽7), C3×C3⋊SO(3,𝔽7) |
| maximal normal subgroup of | D3×SO(3,𝔽7), C3×C3⋊SO(3,𝔽7) |
| maximal quotient of | #72275713, C3×C3⋊SO(3,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 881);