Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C2×C29×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C2×D4 |
| commutator | C4 | C23×C4×C29 |
| Frattini, cosocle | C2×C4 | C24×C29 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C2×C42×C29 ◁ G | |
480 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 3712 |
A minimal faithful permutation representation — 4 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);(3,4,6,5)(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21);(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(3,5,6,4)(8,13)(12,21)(14,18)(15,22)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9088, C29×#9092, C29×#12648, … |
| maximal normal subgroup of | C29×#9088, C29×#9092, C29×#12648, … |
| maximal quotient of | C2×C29×#1579, C2×C29×#1587, C2×C29×#1593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1673);