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×#60996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3198 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1686 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)(26,31)(27,38)(28,33)(29,42)(30,34)(32,39)(35,50)(36,45)(37,46)(40,48)(41,47)(43,49)(44,53)(51,54)(52,55);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26)(25,31)(27,35)(28,41)(29,36)(30,37)(32,44)(33,47)(34,46)(38,50)(39,53)(40,52)(42,45)(43,51)(48,55)(49,54);(10,17)(13,16)(18,23)(20,22)(24,27)(25,32)(26,35)(28,39)(29,43)(30,40)(31,44)(33,48)(34,49)(36,51)(37,52)(38,42)(41,53)(45,50)(46,54)(47,55);(9,14)(13,16)(15,21)(20,22)(24,28,30,29)(25,33,34,42)(26,36,37,41)(27,39,40,43)(31,45,46,47)(32,48,49,38)(35,51,52,53)(44,50,54,55);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(26,37)(31,46)(35,52)(36,41)(44,54)(45,47)(50,55)(51,53)343 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53984);