Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13406 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #2034 |
| prime radical | C7×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
390 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)(10,21)(11,27)(12,19)(13,35)(14,20)(15,23)(16,30)(17,34)(18,33)(22,53)(24,40)(25,42)(26,41)(28,61)(29,62)(31,48)(32,60)(36,47)(37,46)(38,49)(39,58)(43,69)(44,65)(45,66)(50,64)(51,55)(52,59)(54,71)(56,70)(57,68)(63,67);(8,10)(9,15)(11,28)(12,25)(13,36)(14,26)(16,43)(17,41)(18,47)(19,21)(20,42)(22,31)(23,34)(24,38)(27,53)(29,56)(30,63)(32,54)(33,57)(35,52)(37,55)(39,44)(40,50)(45,51)(46,67)(48,71)(49,68)(58,66)(59,64)(60,70)(61,62)(65,69);(8,11,14,32)(9,16,20,45)(10,22)(12,31,34,29)(13,37,38,65)(15,39)(17,46,19,44)(18,48,50,62)(21,51)(23,28)(24,58)(25,54)(26,56)(27,49,60,35)(30,64,66,33)(36,63)(40,53)(41,43)(42,67)(47,70)(52,71)(55,59)(57,69)(61,68)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6394);