Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#25141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,19,16,20,15,25)(14,21,24,27,23,18,22,26);(9,11)(12,14,17,24,16,23,15,22)(13,18,19,26,20,21,25,27);(12,13)(14,21)(15,19)(16,20)(17,25)(18,23)(22,27)(24,26);(9,11)(12,15,16,17)(13,19,20,25)(14,22,23,24)(18,26,21,27)301 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 18129);