Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C4 |
| direct product | C29×#1922 |
| 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 | C23×C29 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C22×C8×C29 ◁ G | |
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 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) = 53 |
| transitive degree | μt(G) = 928 |
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)(30,31)(32,35)(33,34)(36,37)(38,39,42,46,43,41,50,45)(40,44,48,52,49,47,53,51);(31,34)(35,37)(38,40,42,48,43,49,50,53)(39,44,46,52,41,47,45,51);(30,32,33,36)(31,35,34,37)(38,41)(39,43)(40,47)(42,45)(44,49)(46,50)(48,51)(52,53)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C29×#1922, C29×#21041, C29×#21042, … |
| maximal normal subgroup of | C2×C29×#1922, C29×#21041, C29×#21042, … |
| maximal quotient of | C29×#9533, C29×#9534, C29×#9535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 654);