Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#1929 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C22×C29 | C4×D4 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C22×C8×C29 ◁ G | |
224 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 49 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)(31,32)(34,35,38,42,39,37,46,41)(36,40,44,48,45,43,49,47);(30,31,33,32)(34,36,38,44,39,45,46,49)(35,40,42,48,37,43,41,47);(30,32,33,31)(34,37)(35,39)(36,43)(38,41)(40,45)(42,46)(44,47)(48,49)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C29×#1929, C29×#21070, C29×#21071, … |
| maximal normal subgroup of | C2×C29×#1929, C29×#21070, C29×#21071, … |
| maximal quotient of | C29×#9541, C29×#9542, C29×#9543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 661);