Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C29×#2327 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C24 |
| commutator | C22 | C23×C4×C29 |
| Frattini, cosocle | C23 | C24×C29 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 516 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) = 53 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 4 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)(31,33)(34,35,37,36)(39,43)(40,46)(41,49)(51,53);(34,36,37,35)(38,39)(40,47)(41,45)(42,43)(44,46)(48,53)(49,50)(51,52);(30,31,32,33)(38,40)(39,44)(41,48)(42,46)(43,47)(45,51)(49,52)(50,53);(38,41,42,49)(39,45,43,50)(40,48,46,52)(44,51,47,53)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9100, C29×#9104, C29×#9296, … |
| maximal normal subgroup of | C29×#9100, C29×#9104, C29×#9296, … |
| maximal quotient of | C29×#13779, C29×#13786, C29×#13790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1059);