Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C37 · Schur multiplier (H2) = C23 |
| direct product | D4×C16×C37 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C37 | C22 |
| commutator | C2 | C22×C16×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C22×C37 | C22×C8 |
| prime radical | C2×D4×C37 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C37 ◁ G | |
174 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(2,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)2960 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×D4⋊C32, C16×D8×C37, C37×#12038, … |
| maximal normal subgroup of | C37×D4⋊C32, C16×D8×C37, C37×#12038, … |
| maximal quotient of | C16×D8×C37, C16×SD16×C37, C16×Q16×C37, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 899);