Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C37 |
| prime radical | C2×D4×C37 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ G | |
212 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 150 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)(40,42)(41,44)(43,45)(48,52)(50,53);(38,40,43,41)(39,44,45,42)(46,47,49,51)(48,50,52,53);(38,41,43,40)(39,44,45,42)(46,48)(47,50)(49,52)(51,53);(38,42,43,44)(39,40,45,41)925 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#1628, C37×#1638, C37×#1650, … |
| maximal normal subgroup of | C37×#1628, C37×#1638, C37×#1650, … |
| maximal quotient of | C37×#2347, C37×#2354, C37×#2355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 237);