Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#1816 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #121 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
264 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 49 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)(30,31)(32,33)(34,35,39,42,40,37,38,48)(36,43,46,44,47,49,45,41);(34,36,39,46,40,47,38,45)(35,41,48,49,37,44,42,43);(31,33)(34,37)(35,40)(36,44)(38,48)(39,42)(41,47)(43,46)(45,49)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12748, C29×#12749, C29×#12842, … |
| maximal normal subgroup of | C29×#12748, C29×#12749, C29×#12842, … |
| maximal quotient of | C29×#8605, C29×#8606, C29×#8627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 548);