Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13620 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D16 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×#357 |
| prime radical | C2×C7×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
290 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)(8,9,12,17,14,20,33,19)(10,21,24,51,22,50,26,57)(11,27,30,58,32,60,65,59)(13,34,29,18,36,46,64,48)(15,37,39,52,38,54,49,23)(16,41,43,66,45,68,44,31)(25,56,55,69,53,70,35,42)(28,47,62,67,61,71,63,40);(8,10)(9,15)(11,28)(12,24)(13,35)(14,22)(16,42)(17,39)(18,47)(19,49)(20,38)(21,27)(23,31)(25,29)(26,33)(30,62)(32,61)(34,40)(36,55)(37,41)(43,56)(44,70)(45,69)(46,67)(48,71)(50,60)(51,58)(52,66)(53,64)(54,68)(57,59)(63,65);(10,22)(15,38)(21,50)(23,52)(24,26)(25,53)(28,61)(35,55)(37,54)(39,49)(40,67)(42,69)(47,71)(51,57)(56,70)(62,63)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6608);