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 | C2×C7×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D4 |
| commutator | C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C22×C7 | C4⋊C4 |
| prime radical | C23×C7 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(3,4,5,6,7,8,9);(10,11,14,17,15,18,13,23)(12,19,21,24,22,25,20,16);(10,12,13,20,15,22,14,21)(11,16,17,19,18,24,23,25)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 837);