Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#24465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(12,13,17,14,18,20,16,22)(15,23,24,27,25,21,26,19);(8,9)(10,11)(12,14)(13,16)(15,21)(17,20)(18,22)(19,24)(23,25)(26,27);(9,11)(12,15)(13,19)(14,21)(16,24)(17,26)(18,25)(20,27)(22,23);(12,13,18,20)(14,16,22,17)(15,19,25,27)(21,24,23,26)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 17453);