Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C26 |
| direct product | C29×#2614 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| prime radical | C22×D4×C29 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
782 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 1856 |
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)(35,37)(38,39,42,45)(40,46,48,43)(41,44,50,49)(47,52,53,51);(31,33)(35,37)(38,40,42,48)(39,43,45,46)(41,47,50,53)(44,51,49,52);(30,31)(32,33)(34,35)(36,37)(38,39)(40,46)(41,49)(42,45)(43,48)(44,50)(47,51)(52,53);(38,41)(39,44)(40,47)(42,50)(43,51)(45,49)(46,52)(48,53)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#18225, C29×#18227, C29×#23905, … |
| maximal normal subgroup of | C29×#18225, C29×#18227, C29×#23905, … |
| maximal quotient of | C29×#14256, C29×#14263, C29×#14301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1346);