Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C25 |
| direct product | C2×#137×C37 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C37 |
| prime radical | C2×#41×C37 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ G | |
258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(3,4)(5,11)(6,14)(7,9)(8,12)(10,16)(13,18)(15,17);(3,5,7,13)(4,8,9,17)(6,15,16,12)(10,18,14,11);(4,9)(8,17)(10,14)(11,18)814 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#1882, C37×#1902, C37×#1905, … |
| maximal normal subgroup of | C37×#1882, C37×#1902, C37×#1905, … |
| maximal quotient of | C2×C37×#419, C2×C37×#417, C2×C37×#428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 255);