Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,13,17,21)(11,18,19,14)(12,15)(16,20)(22,23)(24,25)(26,35)(27,39,42,49)(28,43)(29,33)(30,34)(31,37)(32,41,48,40)(36,51,53,45)(38,46)(44,47)(50,52,55,54);(8,10)(9,13,15,21)(11,16)(12,17)(14,20,18,23)(19,22)(24,26,28,37,30,38,29,44)(25,31,33,35,34,47,43,46)(27,36,41,54,42,53,40,52)(32,45,49,50,48,51,39,55);(8,11)(9,14)(10,16)(12,19)(13,20)(15,18)(17,22)(21,23)(24,27)(25,32)(26,36)(28,41)(29,40)(30,42)(31,45)(33,49)(34,48)(35,50)(37,54)(38,53)(39,43)(44,52)(46,55)(47,51)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2319);