Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 61 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)(30,31,36,41)(32,42,45,57)(33,46,50,55)(34,40,35,39)(37,51,53,44)(38,48,56,49)(43,54,47,52)(58,61,60,59);(30,32)(31,37)(33,47)(34,51)(35,44)(36,45)(38,54)(39,57)(40,42)(41,53)(43,50)(46,60)(48,59)(49,61)(52,56)(55,58);(30,33,34,48,36,50,35,49)(31,38,39,55,41,56,40,46)(32,43,44,59,45,47,51,61)(37,52,42,58,53,54,57,60)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2049, C29×#23525, C29×#24295, … |
| maximal normal subgroup of | C2×C29×#2049, C29×#23525, C29×#24295, … |
| maximal quotient of | C29×#10444, C29×#10445, C29×#10446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 781);