Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | D4 |
| commutator | C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C2×C7 | C4×D4 |
| prime radical | C7×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×C8 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,6,9,7,10,5,3)(4,8,13,15,14,16,12,11)(17,18,19,20,21,22,23);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(1,4,6,13,7,14,5,12)(2,8,9,15,10,16,3,11)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 851);