Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1571 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| socle | C22×C29 | C2×C4⋊C4 |
| prime radical | C23×C29 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
168 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 1856 |
A minimal faithful permutation representation — 3 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)(31,34)(32,35)(36,37)(38,39,41,44,42,45,48,51)(40,43,47,50,46,49,52,53);(30,31,35,37,33,36,32,34)(38,40)(39,43)(41,47)(42,46)(44,50)(45,49)(48,52)(51,53);(40,46)(43,49)(47,52)(50,53)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8861, C29×#8863, C29×#8878, … |
| maximal normal subgroup of | C29×#8861, C29×#8863, C29×#8878, … |
| maximal quotient of | C29×#7877, C29×#8278, C29×#8599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 303);