Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22×C4 |
| direct product | C37×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | C2×#99 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C2×D4 | C23×C37 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C2×C4×C37 ◁ C2×#41×C37 ◁ G | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 84 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)(40,48,51,63)(41,45)(42,55,57,46)(43,47)(44,50)(49,64,52,66)(53,68,69,61)(54,62)(56,59)(58,65)(60,67);(38,40,43,51)(39,44,47,59)(41,52,54,49)(42,56,57,50)(45,60,62,58)(46,63,55,48)(53,65,69,67)(61,64,68,66);(38,41,43,54)(39,45,47,62)(40,49,51,52)(42,53,57,69)(44,58,59,60)(46,61,55,68)(48,64,63,66)(50,65,56,67)851 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#12712, C37×#12713, C37×#12721, … |
| maximal normal subgroup of | C37×#12712, C37×#12713, C37×#12721, … |
| maximal quotient of | C37×#11434, C37×#11449, C37×#11482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 861);