Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C22 |
| direct product | C2×C37×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C37 | D4 |
| commutator | C4 | C22×C8×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C22×C37 | C4⋊C8 |
| prime radical | C23×C37 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C37 ◁ C22×C8×C37 ◁ G | |
168 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 168 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) = 55 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(3,4,7,12,8,11,16,18,9,13,15,10,6,17,5,14);(3,5,6,15,9,16,8,7)(4,10,11,14,13,12,17,18)2072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7160, C37×#7161, C37×#7162, … |
| maximal normal subgroup of | C37×#7160, C37×#7161, C37×#7162, … |
| maximal quotient of | C2×C37×#1367, C37×#11996, C2×C37×#1370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 884);