Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C29×#2660 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
448 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 448 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) = 49 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 4 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)(34,35)(36,38)(37,39)(40,41)(42,43,46,44)(45,48,49,47);(30,31,32,33)(34,35,37,39)(36,38,41,40)(42,44,46,43)(45,47,49,48);(30,31)(32,33)(34,35)(36,40)(37,39)(38,41)(42,45,46,49)(43,47,44,48);(34,36,37,41)(35,38,39,40)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#10091, C29×#10093, C29×#10099, … |
| maximal normal subgroup of | C29×#10091, C29×#10093, C29×#10099, … |
| maximal quotient of | C29×#14276, C29×#14277, C29×#14282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1392);