Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C25 |
| direct product | C2×#123×C37 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C37 |
| prime radical | C22×D4×C37 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ G | |
342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(4,6)(8,11)(9,13);(7,8)(9,12)(10,11)(13,14);(3,4)(5,6)(7,9,10,13)(8,12,11,14)1036 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#1468, C37×#1890, C37×#1899, … |
| maximal normal subgroup of | C37×#1468, C37×#1890, C37×#1899, … |
| maximal quotient of | C2×C37×#381, C2×C37×#385, C2×C37×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 205);