Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G | |
1640 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)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)(24,25,30,34)(26,31,37,46)(27,39,41,47)(28,33,29,42)(32,43,48,40)(35,49,51,54)(36,45,38,52)(44,53,55,50);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26)(25,31)(27,35)(28,36)(29,38)(30,37)(32,44)(33,45)(34,46)(39,49)(40,50)(41,51)(42,52)(43,53)(47,54)(48,55);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,27)(25,32)(26,35)(28,40)(29,43)(30,41)(31,44)(33,47)(34,48)(36,50)(37,51)(38,53)(39,42)(45,54)(46,55)(49,52);(9,15)(13,16)(14,23)(21,22)(24,28,30,29)(25,33,34,42)(26,36,37,38)(27,40,41,43)(31,45,46,52)(32,47,48,39)(35,50,51,53)(44,54,55,49);(26,37)(31,46)(35,51)(36,38)(44,55)(45,52)(49,54)(50,53)343 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53864);