Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #476 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,24)(15,28,30,35)(16,32)(17,19)(18,23)(20,26,38,42)(21,25)(22,27)(29,41,31,40)(33,37,43,39)(34,36);(8,9,10,11)(12,14,25,34,18,27,16,17)(13,19,32,22,23,36,21,24)(15,26,31,43,30,42,29,33)(20,35,39,40,38,28,37,41);(12,15)(13,20)(14,26)(16,29)(17,33)(18,30)(19,35)(21,37)(22,40)(23,38)(24,41)(25,31)(27,42)(28,36)(32,39)(34,43)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5812);