Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#1690 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C29 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C29 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 3 generators on 53 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)(31,34)(32,36)(35,37)(38,39,42,45)(40,46,48,53)(41,50,47,44)(43,51,52,49);(38,40,47,51,42,48,41,49)(39,43,50,53,45,52,44,46);(30,31,33,35)(32,37,36,34)(40,48)(43,52)(46,53)(49,51)667 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12338, C29×#12339, C29×#12362, … |
| maximal normal subgroup of | C29×#12338, C29×#12339, C29×#12362, … |
| maximal quotient of | C29×#8653, C29×#9228, C29×#9570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 422);