Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#1958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C22×D4×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C2×C42×C29 ◁ G | |
336 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 3 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)(31,33)(34,35,36,37)(38,39)(40,46)(41,49)(42,44)(43,47)(45,50)(48,52)(51,53);(35,37)(38,40,41,47,42,48,50,53)(39,43,45,46,44,51,49,52);(30,31,32,33)(39,44)(43,51)(45,49)(46,52)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7502, C29×#8528, C2×C29×#1958, … |
| maximal normal subgroup of | C29×#7502, C29×#8528, C2×C29×#1958, … |
| maximal quotient of | C29×#9639, C29×#9645, C29×#9681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 690);