Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×SD256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D64 |
| commutator, Frattini, cosocle | C64 | C22×C7 |
| prime radical | C7×D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8 ◁ C7×C16 ◁ C7×C32 ◁ C7×C64 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 2 generators on 135 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(9,71)(10,134)(11,69)(12,132)(13,67)(14,130)(15,65)(16,128)(17,63)(18,126)(19,61)(20,124)(21,59)(22,122)(23,57)(24,120)(25,55)(26,118)(27,53)(28,116)(29,51)(30,114)(31,49)(32,112)(33,47)(34,110)(35,45)(36,108)(37,43)(38,106)(39,41)(40,104)(42,102)(44,100)(46,98)(48,96)(50,94)(52,92)(54,90)(56,88)(58,86)(60,84)(62,82)(64,80)(66,78)(68,76)(70,74)(73,135)(75,133)(77,131)(79,129)(81,127)(83,125)(85,123)(87,121)(89,119)(91,117)(93,115)(95,113)(97,111)(99,109)(101,107)(103,105)469 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 540);