Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#450 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C37 |
| prime radical | C22×D4×C37 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ G | |
188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 592 |
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)(42,43,45,48)(44,46,49,47);(43,46)(44,49)(47,48);(39,41)703 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#1596, C37×#1597, C37×#1599, … |
| maximal normal subgroup of | C37×#1596, C37×#1597, C37×#1599, … |
| maximal quotient of | C37×#1601, C37×#1604, C37×#1613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 131);