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×#61018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ 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)(26,31)(27,37)(28,33)(29,43)(30,34)(32,35)(36,45)(38,47)(39,51)(40,44)(41,48)(42,49)(46,50)(52,55)(53,54);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26,30,29)(25,31,34,43)(27,38,40,35)(28,41,42,36)(32,37,47,44)(33,48,49,45)(39,50,53,52)(46,54,55,51);(10,17)(13,16)(18,23)(20,22)(24,27,30,40)(25,32,34,47)(26,35,29,38)(28,39,42,53)(31,44,43,37)(33,46,49,55)(36,50,41,52)(45,54,48,51);(9,14)(13,16)(15,21)(20,22)(24,28)(25,33)(26,36)(27,39)(29,41)(30,42)(31,45)(32,46)(34,49)(35,50)(37,51)(38,52)(40,53)(43,48)(44,54)(47,55);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(28,42)(33,49)(36,41)(39,53)(45,48)(46,55)(50,52)(51,54)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54006);