Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9369 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
390 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)(10,13,17,23)(11,16)(12,15)(14,18,22,19)(20,21)(24,25)(26,34)(27,38,40,31)(28,41)(29,33)(30,36)(32,43)(35,48,51,55)(37,49)(39,46,42,47)(44,50,54,52)(45,53);(8,10)(9,13,15,23)(11,19)(12,17)(14,21,22,16)(18,20)(24,26,28,36,29,37,43,53)(25,30,32,34,33,45,41,49)(27,35,42,52,40,51,39,50)(31,44,47,48,38,54,46,55);(8,10)(9,14)(11,19)(12,17)(13,21)(15,22)(16,23)(18,20)(24,27)(25,31)(26,35)(28,42)(29,40)(30,44)(32,47)(33,38)(34,48)(36,52)(37,51)(39,43)(41,46)(45,54)(49,55)(50,53)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2357);