Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C29 · Schur multiplier (H2) = C24 |
| direct product | C2×C29×C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C29 | C22 |
| commutator | C2 | C22×C16×C29 |
| Frattini, cosocle | C2×C8 | C23×C29 |
| socle | C23×C29 | C2×C8 |
| prime radical | C24×C29 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C29 ◁ G | |
272 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(3,4)(5,6)(7,8,9,12,10,13,15,18,11,14,16,19,17,20,21,22);(4,6)2320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7183, C29×#7186, C29×#7199, … |
| maximal normal subgroup of | C29×#7183, C29×#7186, C29×#7199, … |
| maximal quotient of | C2×C29×#1314, C2×C29×#1322, C29×#11459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 843);