Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C7 |
| prime radical | C7×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ G | |
104 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,19,20,21,22,23);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 879);