Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2571 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
380 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 4 generators on 53 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)(30,31,34,32)(33,36,37,35)(39,41)(43,44)(46,47,50,53)(48,52,49,51);(31,32)(35,36)(38,39,42,41)(40,43,45,44)(46,48,50,49)(47,51,53,52);(30,32,34,31)(33,36,37,35)(38,40,42,45)(39,43,41,44)(46,47)(48,51)(49,52)(50,53);(30,33,34,37)(31,35,32,36)(38,41,42,39)(40,44,45,43)(46,49,50,48)(47,52,53,51)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#10351, C29×#10352, C29×#10353, … |
| maximal normal subgroup of | C29×#10351, C29×#10352, C29×#10353, … |
| maximal quotient of | C29×#13911, C29×#13982, C29×#14030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1303);