Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#23406 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
2022 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)(14,22)(15,24)(16,19)(17,20)(18,21)(23,26)(25,27);(8,9)(10,11)(12,14)(13,15)(16,23)(17,24)(18,25)(19,22)(20,27)(21,26);(9,11)(12,15)(13,14)(16,22)(17,27)(18,26)(19,23)(20,24)(21,25);(12,16,18,17)(13,19,21,20)(14,23,25,24)(15,22,26,27)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16394);