Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C37 | C2×D4 |
| commutator | C4 | C22×C8×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C2×C37 | C8×D4 |
| prime radical | #135×C37 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C37 ◁ C2×C16×C37 ◁ G | |
144 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39,43,49,44,48,61,67,45,50,62,52,42,58,60,68)(40,51,54,56,53,46,63,57,47,64,66,55,59,65,69,41);(38,40)(39,46)(41,52)(42,59)(43,54)(44,53)(45,47)(48,64)(49,57)(50,65)(51,58)(55,67)(56,68)(60,69)(61,63)(62,66);(38,41,43,51,44,56,61,46,45,57,62,64,42,55,60,65)(39,47,49,66,48,59,67,69,50,40,52,54,58,53,68,63)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12068, C37×#12465, C37×#12466, … |
| maximal normal subgroup of | C37×#12068, C37×#12465, C37×#12466, … |
| maximal quotient of | C37×#12038, C37×#12049, C37×#12050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 903);