Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#435 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C23 |
| commutator | C22 | C22×C4×C37 |
| Frattini, cosocle | C2×C4 | C23×C37 |
| socle | C22×C37 | C22×C4 |
| prime radical | C2×D4×C37 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ G | |
124 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 49 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)(42,43,45,44,46,47,49,48);(39,41);(42,44,45,47,46,48,49,43)1036 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#435, C37×#2929, C37×#2930, … |
| maximal normal subgroup of | C2×C37×#435, C37×#2929, C37×#2930, … |
| maximal quotient of | C37×#1581, C37×#1582, C37×#1583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 116);