Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3197 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)(24,25,29,33)(26,34,35,47)(27,31,38,45)(28,40,42,32)(30,41,43,37)(36,48,49,55)(39,52,51,46)(44,53,54,50);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(26,35)(30,43)(34,47)(36,49)(37,41)(44,54)(48,55)(50,53);(10,17)(13,16)(18,23)(20,22)(24,26,29,35)(25,30,33,43)(27,36,38,49)(28,41,42,37)(31,44,45,54)(32,47,40,34)(39,53,51,50)(46,55,52,48);(9,14)(13,16)(15,21)(20,22)(24,27)(25,31)(26,36)(28,39)(29,38)(30,44)(32,46)(33,45)(34,48)(35,49)(37,50)(40,52)(41,53)(42,51)(43,54)(47,55);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(27,38)(31,45)(36,49)(39,51)(44,54)(46,52)(48,55)(50,53)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53993);