Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2822 |
| 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 | |
380 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 4 generators on 57 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)(35,36)(39,40)(42,43,47,49)(44,50,52,45)(46,48,56,57)(51,55,54,53);(31,33)(34,35,38,36)(37,39,41,40)(42,44,47,52)(43,45,49,50)(46,54,56,51)(48,55,57,53);(30,31)(32,33)(34,36,38,35)(37,39,41,40)(42,45)(43,44)(46,55)(47,50)(48,54)(49,52)(51,57)(53,56);(34,37,38,41)(35,39,36,40)(42,46,47,56)(43,48,49,57)(44,51,52,54)(45,53,50,55)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#10080, C29×#10084, C29×#10087, … |
| maximal normal subgroup of | C29×#10080, C29×#10084, C29×#10087, … |
| maximal quotient of | C29×#14629, C29×#14631, C29×#14634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1554);