Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C37 | C22⋊C4 |
| commutator | C22 | C22×C8×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C2×C37 | C2×C22⋊C8 |
| prime radical | C2×#41×C37 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C37 ◁ C22×C8×C37 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 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) = 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)(38,39,41,46,42,47,53,61,43,45,54,58,55,59,66,68)(40,48,50,60,51,62,64,69,49,44,52,56,63,57,65,67);(38,40)(39,44)(41,52)(42,51)(43,49)(45,48)(46,60)(47,57)(50,54)(53,65)(55,63)(56,58)(59,62)(61,69)(64,66)(67,68);(40,49)(44,48)(50,52)(51,63)(56,60)(57,62)(64,65)(67,69)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7207, C37×#7208, C37×#11483, … |
| maximal normal subgroup of | C37×#7207, C37×#7208, C37×#11483, … |
| maximal quotient of | C37×#11419, C37×#11441, C37×#11461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 847);