Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2980 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C24 |
| commutator | C22 | C23×C4×C29 |
| Frattini, cosocle | C2×C4 | C24×C29 |
| socle | C22×C29 | C23×C4 |
| prime radical | C29×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ G | |
394 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 248 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)(32,36)(33,35)(34,37)(38,39,41,45,42,46,49,53)(40,43,48,50,47,44,51,52);(30,32)(31,34)(33,37)(35,36)(38,40)(39,43)(41,48)(42,47)(44,46)(45,50)(49,51)(52,53);(32,37)(34,36)(38,40)(39,44)(41,48)(42,47)(43,46)(45,52)(49,51)(50,53);(31,35)(34,36)(40,47)(43,44)(48,51)(50,52)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8918, C29×#9005, C29×#9007, … |
| maximal normal subgroup of | C29×#8918, C29×#9005, C29×#9007, … |
| maximal quotient of | C29×#20532, C29×#20539, C29×#20563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1712);