Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C25 |
| direct product | C37×#536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C37 |
| prime radical | C2×#41×C37 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ G | |
270 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 270 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) = 57 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 4 generators on 57 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)(39,41)(43,47)(44,50)(45,53)(55,57);(42,43,46,47)(44,51,50,48)(45,49,53,54)(52,57,56,55);(38,39)(40,41)(42,44,46,50)(43,48,47,51)(45,52,53,56)(49,55,54,57);(42,45)(43,49)(44,52)(46,53)(47,54)(48,55)(50,56)(51,57)814 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#1909, C37×#1912, C37×#1915, … |
| maximal normal subgroup of | C37×#1909, C37×#1912, C37×#1915, … |
| maximal quotient of | C37×#2310, C37×#2311, C37×#2317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 217);