Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#19226 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,17)(11,16)(12,14)(13,20)(15,18)(19,23)(21,22)(24,25,29,33)(26,34,38,48)(27,31,39,45)(28,41,35,32)(30,42,44,37)(36,47,50,55)(40,52,49,46)(43,53,54,51);(9,14)(15,22)(16,20)(17,23)(24,26,35,42,29,38,28,37)(25,30,41,48,33,44,32,34)(27,36,49,53,39,50,40,51)(31,43,52,55,45,54,46,47);(8,10)(9,15)(11,18)(12,19)(13,21)(14,22)(16,23)(17,20)(24,27)(25,31)(26,36)(28,40)(29,39)(30,43)(32,46)(33,45)(34,47)(35,49)(37,51)(38,50)(41,52)(42,53)(44,54)(48,55);(8,11,12,13)(9,16,14,20)(10,18,19,21)(15,23,22,17)(27,39)(31,45)(36,50)(40,49)(43,54)(46,52)(47,55)(51,53)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12214);