Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#1403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C4×C29 |
| Frattini, cosocle | #124 | C22×C29 |
| prime radical | C29×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C29 ◁ C2×C4×C29 ◁ C29×#124 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 2 generators on 45 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)(30,31,33,37,35,32,39,41)(34,42,44,40,45,38,43,36);(30,32)(31,35)(33,40)(34,38)(36,39)(37,43)(41,44)(42,45)493 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1403, C29×#13564, C29×#13565, … |
| maximal normal subgroup of | C2×C29×#1403, C29×#13564, C29×#13565, … |
| maximal quotient of | C29×#7344, C29×#7375, C29×#7409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 135);