Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C26 |
| direct product | Q8×C22⋊C4×C29 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C24 |
| commutator | C22 | C23×C4×C29 |
| Frattini, cosocle | C23 | C24×C29 |
| prime radical | C24×C29 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
820 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 392 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) = 1856 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4)(5,8,7,6)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(1,5,3,7)(2,6,4,8);(9,10,12,14)(11,13,15,16);(10,13)(14,16)1450 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9258, C29×#9262, C29×#22437, … |
| maximal normal subgroup of | C29×#9258, C29×#9262, C29×#22437, … |
| maximal quotient of | C22⋊C4×C4⋊C4×C29, Q8×C29×#95, C29×#13817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1072);