Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#539 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C37 |
| prime radical | #121×C37 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ G | |
218 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39)(40,41)(42,43)(44,45)(48,52)(50,53);(39,43)(40,44)(46,47,49,51)(48,50,52,53);(38,40,42,44)(39,41,43,45)(46,48)(47,50)(49,52)(51,53);(38,41)(39,44)(40,43)(42,45)814 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#1597, C37×#1603, C37×#1609, … |
| maximal normal subgroup of | C37×#1597, C37×#1603, C37×#1609, … |
| maximal quotient of | C37×#2313, C37×#2318, C37×#2322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 220);