Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | ΓL(1,𝔽8)×#392 |
| Fitting, Hall | C7×#392 ⋊ C3 |
| Sylow | C7 ⋊ C3×#392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C7×#392 | C3 |
| hypercenter | #392 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#392 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #392 | |
644 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3,4)(5,7,6)(13,15)(16,17)(18,19);(1,2,3,5,4,6,7)(9,11)(13,15)(17,19);(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)110 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#392, #72259713, #72259716, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#392, #72259713, #72259716, … |
| maximal quotient of | ΓL(1,𝔽8)×#2002, ΓL(1,𝔽8)×#2008, ΓL(1,𝔽8)×#1438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4212);