Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12806 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #410 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| socle | C2×C7 | #1799 |
| prime radical | C7×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×C8 ◁ C2×C7×#131 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,22)(12,25,26,15)(13,17)(14,20)(16,24)(19,33)(21,31,35,32)(23,30,38,37)(27,39,28,36)(29,34);(8,10,11,20,13,19,24,34)(9,14,16,18,17,29,22,33)(12,21,23,36,26,35,38,39)(15,28,30,32,25,27,37,31);(9,15)(10,19)(14,27)(16,30)(17,25)(18,31)(20,34)(21,35)(22,37)(28,29)(32,33)(36,39)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5794);