Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2582 |
| 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 | |
484 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 248 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)(31,33)(34,35,38,36)(37,40,41,39)(42,43,46,49)(44,48,45,47);(35,36)(39,40)(42,44,46,45)(43,47,49,48);(30,31)(32,33)(34,36,38,35)(37,40,41,39)(42,43)(44,47)(45,48)(46,49);(34,37,38,41)(35,39,36,40)(42,45,46,44)(43,48,49,47)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#10230, C29×#10234, C29×#10237, … |
| maximal normal subgroup of | C29×#10230, C29×#10234, C29×#10237, … |
| maximal quotient of | C29×#13926, C29×#14017, C29×#14041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1314);