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×#2642 |
| 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 200 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 200 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)(30,31,32,33)(36,40)(38,41)(43,47)(48,50)(49,55)(56,57);(30,31,32,33)(34,35,37,39)(36,38,40,41)(42,43)(44,50)(45,49)(46,47)(48,52)(51,57)(53,56)(54,55);(30,31)(32,33)(34,36)(35,38)(37,40)(39,41)(42,44,46,52)(43,48,47,50)(45,53,54,51)(49,57,55,56);(42,45,46,54)(43,49,47,55)(44,51,52,53)(48,56,50,57)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#10095, C29×#10103, C29×#18441, … |
| maximal normal subgroup of | C29×#10095, C29×#10103, C29×#18441, … |
| maximal quotient of | C29×#14261, C29×#14274, C29×#14283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1374);