Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9,12,15,13,16,11,21)(10,17,20,22,18,23,19,14)(24,25)(26,35,38,46)(27,39)(28,33)(29,43)(30,34)(31,37,47,42)(32,40)(36,51,54,45)(41,48)(44,49)(50,52,55,53);(8,10,13,18)(9,14)(11,19,12,20)(15,17)(16,22)(21,23)(24,26,28,42,30,38,29,37)(25,31,33,46,34,47,43,35)(27,36,40,53,41,54,44,52)(32,45,48,55,49,51,39,50);(10,18)(14,22)(17,23)(19,20)(24,27)(25,32)(26,36)(28,40)(29,44)(30,41)(31,45)(33,48)(34,49)(35,50)(37,52)(38,54)(39,43)(42,53)(46,55)(47,51)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2302);