Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C29 · Schur multiplier (H2) = C23 |
| direct product | C2×C29×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C29 | D4 |
| commutator | C4 | C22×C8×C29 |
| Frattini, cosocle | C2×C8 | C23×C29 |
| socle | C22×C29 | C22⋊C8 |
| prime radical | C2×#41×C29 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C29 ◁ C22×C8×C29 ◁ G | |
224 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 224 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) = 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);(3,4,7,12,8,13,22,30,9,14,23,31,6,11,20,28)(5,15,17,32,16,26,33,29,18,27,21,34,19,10,24,25);(3,5)(4,10)(6,19)(7,21)(8,16)(9,18)(11,27)(12,29)(13,15)(14,26)(17,23)(20,33)(22,24)(25,31)(28,32)(30,34)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×C29×#350, C29×#12604, C29×#12614, … |
| maximal normal subgroup of | C4×C29×#350, C29×#12604, C29×#12614, … |
| maximal quotient of | C2×C29×#1315, C29×#11422, C2×C29×#1323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 848);