Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C2×C29×#359 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | D8 |
| commutator | C8 | C22×C4×C29 |
| Frattini, cosocle | C2×C8 | C23×C29 |
| socle | C22×C29 | #102 |
| prime radical | C2×C29×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C22×C4×C29 ◁ C22×C8×C29 ◁ G | |
224 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 224 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) = 63 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 63 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63);(3,4,7,12,9,13,8,23)(5,14,16,30,17,31,24,28)(6,18,20,29,22,33,21,11)(10,19,25,32,27,34,26,15);(3,5)(4,10)(6,19)(7,16)(8,24)(9,17)(11,28)(12,25)(13,27)(14,18)(15,21)(20,32)(22,34)(23,26)(29,30)(31,33)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×C29×#359, C29×#12640, C29×#12641, … |
| maximal normal subgroup of | C4×C29×#359, C29×#12640, C29×#12641, … |
| maximal quotient of | C2×C29×#1331, C2×C29×#1332, C29×#11591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 874);