Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of C43⋊C7 |
| automorphism | ? |
| homology | Abelianization (H1) = C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C43⋊C7 |
| Fitting, derived, Sylow | C43 ⋊ C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C43 | C7 |
| Frattini, socle | C23 | AGL(1,𝔽8) |
| LCS | G ▷ C43 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,5,9,14,18,22)(3,7,12,15,19,23,26)(4,6,11,17,21,24,27)(8,13,10,16,20,25,28);(1,3,4,8)(2,7,6,13)(5,12,11,10)(9,17)(15,16)(18,25,24,23)Polycyclic presentation:
7,2^6|b^2=efg,c^2=g,d^2=fg,b^a=c,c^a=d,d^a=bdg,e^a=eg,f^a=efg,g^a=fg16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 178);