Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #121 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #390 |
| prime radical | C2×C29×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 284 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)(32,36,41,50)(33,43)(34,46)(35,38)(37,44)(39,47)(40,52,53,61)(42,56,55,49)(45,57)(48,59,60,54)(51,58);(30,32)(31,36,38,50)(33,40)(34,42)(35,41)(37,48,51,60)(39,49,46,56)(43,52,57,61)(44,54)(45,53)(47,55)(58,59);(30,33,34,44,35,45,47,58)(31,37,39,43,38,51,46,57)(32,40,42,54,41,53,55,59)(36,48,49,52,50,60,56,61)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13016, C29×#13021, C29×#13030, … |
| maximal normal subgroup of | C29×#13016, C29×#13021, C29×#13030, … |
| maximal quotient of | C29×#9667, C29×#9668, C29×#9669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 708);