Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33538 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C2×C7 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 71 points:
(1,2,3,4,5,6,7)(8,9)(10,22)(11,27)(12,18)(13,33)(14,20)(15,21)(16,25)(17,30)(19,35)(23,51)(24,49)(26,50)(28,43)(29,60)(31,47)(32,46)(34,39)(36,45)(37,44)(38,65)(40,64)(41,56)(42,63)(48,54)(52,70)(53,57)(55,68)(58,62)(59,69)(61,67)(66,71);(8,10)(9,16)(11,23)(12,24)(13,34)(14,37)(15,26)(17,38)(18,39)(19,44)(20,40)(21,41)(22,33)(25,35)(27,57)(28,52)(29,61)(30,54)(31,55)(32,56)(36,64)(42,68)(43,67)(45,50)(46,49)(47,66)(48,58)(51,60)(53,62)(59,71)(63,65)(69,70);(8,11)(9,17)(10,23)(12,28)(13,29)(14,30)(15,31)(16,38)(18,27)(19,42)(20,47)(21,43)(22,48)(24,52)(25,53)(26,55)(32,59)(33,58)(34,61)(35,62)(36,63)(37,54)(39,57)(40,66)(41,67)(44,68)(45,69)(46,60)(49,51)(50,70)(56,71)(64,65);(8,12,15,14)(9,18,21,20)(10,24,26,37)(11,28,31,30)(13,32,36,35)(16,39,41,40)(17,27,43,47)(19,33,46,45)(22,49,50,44)(23,52,55,54)(25,34,56,64)(29,59,63,62)(38,57,67,66)(42,58,60,69)(48,51,70,68)(53,61,71,65)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26526);