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×#551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C37 |
| prime radical | #120×C37 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ G | |
248 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 248 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) = 592 |
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)(38,39,41,45)(40,46,47,43)(42,44,49,51)(48,52,53,50);(38,40,42,48)(39,43,44,50)(41,47,49,53)(45,46,51,52);(39,44)(43,50)(45,51)(46,52);(40,47)(43,46)(48,53)(50,52)925 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#2197, C2×C37×#551, C37×#3475, … |
| maximal normal subgroup of | C37×#2197, C2×C37×#551, C37×#3475, … |
| maximal quotient of | C37×#2342, C37×#2343, C37×#2353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 232);