Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #386 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
314 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 108 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)(30,31,35,39,36,40,34,50)(32,41,45,52,44,56,43,37)(33,46,48,54,49,38,47,53)(42,55,59,51,58,61,57,60);(30,32)(31,37,40,52)(33,42)(34,43)(35,45)(36,44)(38,51,46,60)(39,41,50,56)(47,57)(48,59)(49,58)(53,61,54,55);(30,33,36,49)(31,38,40,46)(32,42,44,58)(34,48,35,47)(37,51,52,60)(39,54,50,53)(41,55,56,61)(43,59,45,57)841 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12781, C29×#12782, C2×C29×#1912, … |
| maximal normal subgroup of | C29×#12781, C29×#12782, C2×C29×#1912, … |
| maximal quotient of | C29×#9100, C29×#9102, C29×#9104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 644);