Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2634 |
| 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 | |
416 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 192 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)(35,38)(39,41)(42,43,47,45)(44,50,52,48)(46,49,54,53)(51,56,57,55);(31,33)(35,38)(39,41)(42,44,47,52)(43,48,45,50)(46,51,54,57)(49,55,53,56);(30,31)(32,33)(34,35)(36,39)(37,38)(40,41)(42,45,47,43)(44,48,52,50)(46,49,54,53)(51,56,57,55);(34,36,37,40)(35,39,38,41)(42,46)(43,49)(44,51)(45,53)(47,54)(48,55)(50,56)(52,57)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#9190, C29×#10021, C29×#10025, … |
| maximal normal subgroup of | C29×#9190, C29×#10021, C29×#10025, … |
| maximal quotient of | C29×#14270, C29×#14289, C29×#14298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1366);