Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#11923460 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | S42 |
| commutator, cosocle | C2×A42 | C23 |
| Frattini | C2 | C2×S42 |
| Fitting, socle | C26 | D32 |
| hypocenter | A42 | C2×D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C22 | |
3225 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 3225 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(7,8,10)(11,14,12);(4,6)(8,9)(13,14);(3,4,5,6)(13,14)76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215293, D3×#11923460, #72259050 |
| maximal normal subgroup of | C2×#72215293, #72259050 |
| maximal quotient of | #72226288, #72226315, #72226321, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(7,8,10)(11,14,12), (4,6)(8,9)(13,14), (3,4,5,6)(13,14));