Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×SO(3,𝔽7) |
| commutator, cosocle, hypocenter | SL(2,𝔽7) | C22 |
| Fitting, hypercenter | C4 | SO(3,𝔽7) |
| prime radical | #10621465 | C2 |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C2 ◁ C4 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,42,7,38,2,40,4,45,3,41,5,44)(6,36,12,46,11,34,9,33,8,48,15,37)(10,39,13,35)(14,43,16,47)(17,58,23,54,18,56,20,61,19,57,21,60)(22,52,28,62,27,50,25,49,24,64,31,53)(26,55,29,51)(30,59,32,63);(1,56,5,55,8,62,15,63,4,60,2,51,12,64,11,59)(3,54,14,49,6,50,13,61,7,57,16,52,9,53,10,58)(17,44,21,35,24,48,31,43,20,40,18,39,28,46,27,47)(19,41,30,36,22,37,29,42,23,38,32,33,25,34,26,45)23 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11292);