Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12829 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #418 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1804 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
378 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)(10,20,23,15)(11,25)(12,29)(13,35)(14,17)(16,27)(18,33)(19,32)(21,48,49,67)(22,52,30,63)(24,39,55,47)(26,58)(28,61)(31,45)(34,62)(36,44)(37,50,46,60)(38,56,43,65)(40,66)(41,59)(42,53)(51,69,57,71)(54,68,64,70);(8,10,13,24,14,23,32,55)(9,15,35,47,17,20,19,39)(11,21,36,57,26,49,28,51)(12,30,33,65,34,22,53,56)(16,37,40,68,41,46,45,70)(18,43,62,63,42,38,29,52)(25,48,44,71,58,67,61,69)(27,60,66,54,59,50,31,64);(8,11)(9,16)(10,21)(12,31)(13,36)(14,26)(15,37)(17,41)(18,44)(19,45)(20,46)(22,54)(23,49)(24,57)(25,29)(27,33)(28,32)(30,64)(34,66)(35,40)(38,69)(39,70)(42,61)(43,71)(47,68)(48,52)(50,56)(51,55)(53,59)(58,62)(60,65)(63,67)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5817);