Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C2×C29×#328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C24×C29 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C23×C4×C29 ◁ G | |
336 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 336 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) = 47 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(4,8,9,17)(5,12)(10,18,11,13)(14,16);(3,4,6,10)(5,13,12,17)(7,11,15,9)(8,16,18,14)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×C29×#328, C29×#8217, C29×#8219, … |
| maximal normal subgroup of | C4×C29×#328, C29×#8217, C29×#8219, … |
| maximal quotient of | C2×C29×#1271, C2×C29×#1272, C29×#7566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 231);