Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, hypercenter | C4 | SO(3,𝔽7) |
| commutator, hypocenter | SL(2,𝔽7) | C4 |
| socle | C2 | GL(3,𝔽2)⋊C4 |
| prime radical, cosocle | #10621465 | C2 |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C4 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,6,16,10,11,4,5,7,9,14,13,15)(8,12)(17,18,19,22,32,26,27,20,21,23,25,30,29,31)(24,28)(33,37,46,44,43,39,38,36,34,48,40,47,35,41)(42,45)(49,53,62,60,59,55,54,52,50,64,56,63,51,57)(58,61);(1,38,25,55,8,46,32,53,4,41,22,51,12,48,30,50)(2,33,27,56,13,45,19,59,5,36,31,60,10,42,23,63)(3,43,18,49,15,44,29,61,7,47,21,52,11,40,26,58)(6,35,24,62,14,34,20,57,9,39,28,64,16,37,17,54)32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6312);