Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1945 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C22×C29 | C4×D4 |
| prime radical | C23×C29 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C22×C8×C29 ◁ G | |
208 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 928 |
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)(34,35,39,43,40,44,38,48)(36,41,45,49,46,42,37,47);(31,33)(34,36,39,45,40,46,38,37)(35,41,43,49,44,42,48,47);(34,37,38,46,40,45,39,36)(35,42,43,47,44,41,48,49)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7119, C29×#12939, C29×#12951, … |
| maximal normal subgroup of | C29×#7119, C29×#12939, C29×#12951, … |
| maximal quotient of | C29×#9593, C29×#9594, C29×#9599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 677);