Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#7538 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C7 | D32 |
| commutator | C32 | C2×C4×C7 |
| Frattini, cosocle | C2×C32 | C22×C7 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C2×C7×C16 ◁ C2×C7×C32 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4,5,6,7)(8,9,10,11)(13,19)(14,24)(15,26)(16,31)(17,35)(20,42)(21,46)(22,49)(23,41)(25,30)(27,57)(28,61)(29,38)(32,66)(33,43)(34,39)(36,47)(37,40)(44,74)(45,73)(48,75)(50,51)(52,59)(53,67)(54,65)(55,56)(58,63)(60,71)(62,64)(68,72)(69,70);(12,13,18,23)(14,25,29,54)(15,30,33,65)(16,34,36,68)(17,37,35,22)(19,38,41,24)(20,43,45,26)(21,47,48,31)(27,58,60,74)(28,62,59,53)(32,67,55,64)(39,71,72,57)(40,52,49,61)(42,69,73,50)(44,56,63,66)(46,70,75,51)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 526);