Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C8×#6862 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | H(3,𝔽3) |
| commutator | C33 | C32×C8 |
| Frattini, cosocle | C33×C4 | C2×C32 |
| socle | C2×C32 | C4×H(3,𝔽3) |
| prime radical | C2×C33 | C32×C4 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ C33×C8 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 62 points:
(1,2,3,4,5,6,7,8)(9,10,13)(11,17,21)(12,16,22)(14,23,19)(15,26,20)(18,28,31)(24,34,35)(25,33,30)(27,32,29)(36,37,40,47,53,59,39,42,49)(38,43,54,45,52,48,46,41,50)(44,51,58,57,55,60,56,61,62);(9,11,18,12,19,29,20,30,35)(10,14,24,16,25,28,15,17,27)(13,21,31,22,23,32,26,33,34)(36,38,44)(37,41,51)(39,46,56)(40,48,58)(42,52,61)(43,55,53)(45,57,47)(49,54,62)(50,60,59)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 78);