Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×#12056 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C8×D4 |
| prime radical | C2×C7×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C16 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,20,23,21,24,28,30,19,18,22,25,27,26,29,31);(9,12)(10,15)(13,14)(17,18)(23,25)(24,26)(30,31);(9,13)(12,14)(16,18,20,25,21,26,28,31,19,17,22,23,27,24,29,30)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5044);