Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C13 · Schur multiplier (H2) = C22 |
| direct product | C3×C13×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C13 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C3×C13 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C13 ◁ C2×C3×C4×C13 ◁ G | |
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 312 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18)(19,21)(20,22)(23,24);(18,21)(19,24)(22,23);(18,22)(21,23)429 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 1150);