Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2120 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #99 |
| commutator | C23 | C22×C4×C29 |
| Frattini, cosocle | C2×D4 | C23×C29 |
| socle | C2×C29 | C2×#99 |
| prime radical | C29×#525 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C2×C4×C29 ◁ C2×#41×C29 ◁ G | |
264 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 3 generators on 45 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)(33,40,41,35)(34,37)(38,44,45,42)(39,43);(31,35)(36,42)(37,40)(43,44);(30,32,34,39)(31,36,37,43)(33,38,41,45)(35,42,40,44)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12623, C29×#12624, C29×#12711, … |
| maximal normal subgroup of | C29×#12623, C29×#12624, C29×#12711, … |
| maximal quotient of | C29×#11425, C29×#11452, C29×#11471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 852);