Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C2×C29×#360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #102 |
| commutator | C8 | C22×C4×C29 |
| Frattini, cosocle | C2×C8 | C23×C29 |
| prime radical | C2×C29×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ C22×C8×C29 ◁ G | |
232 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(4,8,9,17)(5,11)(6,14)(10,12,15,18)(13,16);(3,4)(5,12)(6,15)(7,9)(8,16)(10,14)(11,17)(13,18)928 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C29×#360, C29×#12650, C29×#12735, … |
| maximal normal subgroup of | C4×C29×#360, C29×#12650, C29×#12735, … |
| maximal quotient of | C2×C29×#1333, C2×C29×#1334, C29×#11598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 876);