Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #102 |
| commutator | C8 | C42×C29 |
| Frattini, cosocle | C22×C8 | C22×C29 |
| prime radical | C23×C29 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ C22×C8×C29 ◁ G | |
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 2 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)(35,39)(36,43)(37,44)(40,42)(41,48)(45,49)(46,47);(34,35,36,40,44,46,49,47,38,42,45,39,37,41,43,48)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1387, C29×#12569, C4×C29×#361, … |
| maximal normal subgroup of | C2×C29×#1387, C29×#12569, C4×C29×#361, … |
| maximal quotient of | C29×#7159, C29×#7176, C29×#7178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 119);