Finite Group DatabasebetaFinite Group Databasebeta| order | 1701 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C7 · Schur multiplier (H2) = C9 |
| direct product | C7×#6898 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C7 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C7 |
| prime radical | C33×C7 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C7 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 567 |
A minimal faithful permutation representation — 3 generators on 61 points:
(1,2,3,4,5,6,7)(8,9,12)(10,14,20)(11,15,19)(13,21,18)(16,23,28)(17,24,27)(22,29,26)(25,31,33)(30,34,32)(35,36,39)(37,40,46)(38,42,47)(41,51,45)(43,48,54)(44,50,55)(49,58,53)(52,57,60)(56,61,59);(9,13,15)(12,19,21)(14,22,24)(20,27,29)(23,30,31)(28,33,34)(35,37,43,38,44,52,45,53,59)(36,40,48,42,50,57,41,49,56)(39,46,54,47,55,60,51,58,61);(8,10,16,18,26,32,11,17,25)(9,14,23,13,22,30,15,24,31)(12,20,28,21,29,34,19,27,33)(36,41,42)(39,47,51)(40,49,50)(46,55,58)(48,56,57)(54,60,61)357 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1701, 45);