Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13392 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #2033 |
| prime radical | C7×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
386 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)(11,26)(12,32)(13,35)(14,19)(15,22)(16,29)(17,34)(18,33)(21,51)(23,40)(24,41)(25,42)(27,61)(28,59)(30,60)(31,47)(36,46)(37,44)(38,65)(39,67)(43,64)(45,63)(48,66)(49,53)(50,56)(52,68)(54,70)(55,69)(57,58)(62,71);(8,10)(9,15)(11,27)(12,24)(13,36)(14,25)(16,39)(17,42)(18,46)(19,41)(20,32)(21,30)(22,34)(23,38)(26,58)(28,52)(29,53)(31,57)(33,54)(35,50)(37,55)(40,48)(43,71)(44,49)(45,69)(47,51)(56,66)(59,61)(60,68)(62,63)(64,67)(65,70);(8,11,12,28,14,30,34,31)(9,16,17,43,19,45,32,44)(10,21,22,52,25,27,24,57)(13,37,33,63,38,64,66,29)(15,39,20,49,41,69,42,71)(18,47,35,60,48,59,65,26)(23,55,54,53,36,67,56,62)(40,61,50,68,46,51,70,58)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6380);