Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×#11921058 |
| Fitting | C2×C3×H(3,𝔽4) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11921058 |
| commutator, cosocle | C3×#120⋊C3 | C23 |
| Frattini | C23 | C22×C3⋊S4 |
| Fitting | C2×C3×H(3,𝔽4) | D3 |
| socle | C23×C3 | GL(2,ℤ/4) |
| hypocenter | C3×C22⋊A4 | C2×D4 |
| LCS | G ▷ C3×#120⋊C3 ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
1300 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 21 points:
(3,4,5)(6,7,17,11)(8,20,18,21)(9,12,10,14)(13,16,19,15);(4,5)(7,8)(9,10)(11,18)(12,16)(14,15)(20,21);(1,2)(7,8,13)(9,10,17)(11,15,21)(12,16,19)(14,18,20)66 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72258621 |
| maximal quotient of | C2×#72258621 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5)(6,7,17,11)(8,20,18,21)(9,12,10,14)(13,16,19,15), (4,5)(7,8)(9,10)(11,18)(12,16)(14,15)(20,21), (1,2)(7,8,13)(9,10,17)(11,15,21)(12,16,19)(14,18,20));