Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,15,21)(10,22,27,55)(11,28,32,52)(12,33,38,50)(13,20,14,19)(16,40,44,24)(17,39,47,31)(18,36,51,37)(23,56,59,71)(25,54,26,48)(29,63,58,53)(30,49,35,46)(34,43,61,42)(41,67,70,64)(45,60,69,57)(62,68,66,65);(8,10)(9,16)(11,29)(12,34)(13,25)(14,26)(15,27)(17,45)(18,48)(19,42)(20,43)(21,44)(22,33)(23,31)(24,37)(28,41)(30,60)(32,58)(35,57)(36,40)(38,61)(39,59)(46,53)(47,69)(49,63)(50,55)(51,54)(52,70)(56,62)(64,65)(66,71)(67,68);(8,11)(9,17)(10,23)(12,35)(13,39)(14,31)(15,32)(16,41)(18,49)(19,52)(20,28)(21,47)(22,53)(24,60)(25,29)(26,58)(27,59)(30,38)(33,66)(34,67)(36,65)(37,68)(40,57)(42,45)(43,69)(44,70)(46,51)(48,71)(50,62)(54,56)(55,63)(61,64)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6312);