Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C22 |
| commutator | C2 | C22×C7×C8 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle, prime radical | C22×C7 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15);(16,17,18,19)(20,23,22,21);(16,20,18,22)(17,21,19,23)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 853);