Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1392 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #99 |
| commutator | C23 | C42×C29 |
| Frattini, cosocle | C2×#41 | C22×C29 |
| socle | C2×C29 | #342 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C2×C4×C29 ◁ C2×#41×C29 ◁ G | |
158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 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)(30,31,34,39,36,41,35,40)(32,42,44,57,46,59,45,58)(33,47,48,56,49,38,50,54)(37,51,52,43,53,60,55,61);(30,32,35,45,36,46,34,44)(31,37,40,52,41,53,39,55)(33,43,48,60,49,61,50,51)(38,42,54,58,47,59,56,57)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7400, C29×#7401, C29×#7407, … |
| maximal normal subgroup of | C29×#7400, C29×#7401, C29×#7407, … |
| maximal quotient of | C29×#7188, C29×#7203, C29×#7216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 124);