Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×C7×C8×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7×C8 | C22 |
| commutator | C2 | C24×C7×C8 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C22×C7×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7×C8 ◁ G | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(3,4,5,6,7,8,9,10);(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(11,13,14,17)(12,16,15,18)1120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53184);