Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#3160 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C29 |
| socle | C22×C29 | C22×D4 |
| prime radical | C23×C29 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
356 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 224 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) = 1856 |
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)(30,31,33,35)(32,36,37,34)(38,39)(40,47)(41,45)(42,51)(43,46)(44,49)(48,52)(50,53);(30,32,33,37)(31,34,35,36);(38,40,42,49,43,50,41,48)(39,44,45,47,46,52,51,53);(38,41,43,42)(39,45,46,51)(40,48,50,49)(44,47,52,53)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9068, C29×#9749, C29×#9752, … |
| maximal normal subgroup of | C29×#9068, C29×#9749, C29×#9752, … |
| maximal quotient of | C29×#20797, C29×#21151, C29×#21318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1892);