Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#19060 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
822 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)(10,16)(11,18)(13,21)(14,19)(17,20)(22,23)(24,25,30,35)(26,36,39,48)(27,40,41,49)(28,34,29,33)(31,43,46,38)(32,44,47,42)(37,45,51,55)(50,52,54,53);(8,9,12,15)(10,13,17,22)(11,19,18,14)(16,23,20,21)(24,26)(25,31)(27,37)(28,43)(29,38)(30,39)(32,45)(33,48)(34,36)(35,46)(40,54)(41,51)(42,53)(44,52)(47,55)(49,50);(8,10)(9,13)(11,20)(12,17)(14,23)(15,22)(16,18)(19,21)(24,27)(25,32)(26,37)(28,42)(29,44)(30,41)(31,45)(33,40)(34,49)(35,47)(36,50)(38,52)(39,51)(43,53)(46,55)(48,54);(27,41)(32,47)(37,51)(40,49)(42,44)(45,55)(50,54)(52,53)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12048);