Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#1458 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C22⋊C4 |
| commutator | C22 | C22×C8×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C22×C29 | C2×C22⋊C4 |
| prime radical | C24×C29 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
264 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 53 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)(30,31,32,34,33,35,36,37)(38,39)(40,46,48,43)(41,44)(42,45)(47,52,53,50)(49,51);(38,40)(39,43)(41,47)(42,48)(44,50)(45,46)(49,53)(51,52);(38,41,42,49)(39,44,45,51)(40,47,48,53)(43,50,46,52)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7188, C29×#7190, C29×#7252, … |
| maximal normal subgroup of | C29×#7188, C29×#7190, C29×#7252, … |
| maximal quotient of | C29×#7560, C29×#7562, C29×#7574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 190);