Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C29 |
| prime radical | C29×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C29 ◁ C2×D4×C29 ◁ G | |
226 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 61 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)(32,40)(33,37)(34,38)(35,39)(36,42)(41,57)(43,56)(44,55)(45,51)(46,52)(47,54)(48,58)(49,59)(50,60)(53,61);(30,32,35,43)(31,36,39,49)(33,44,46,41)(34,42,47,59)(37,50,52,48)(38,40,54,56)(45,60,61,58)(51,55,53,57);(30,33,35,46)(31,37,39,52)(32,41,43,44)(34,45,47,61)(36,48,49,50)(38,53,54,51)(40,55,56,57)(42,58,59,60)493 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13679, C29×#13680, C2×C29×#2201, … |
| maximal normal subgroup of | C29×#13679, C29×#13680, C2×C29×#2201, … |
| maximal quotient of | C29×#12095, C29×#12101, C29×#12125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 933);