Finite Group DatabasebetaFinite Group Databasebeta| order | 10368 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C32 |
| direct product | C2×#72220371 |
| Fitting | C2×C32×H(5,𝔽2) ⋊ C3⋊D3 |
| derived | C3×Ω+(4,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #72213301 |
| commutator, cosocle, hypocenter | C3×Ω+(4,𝔽3) | C22×C3 |
| Frattini | C2×C3 | C2×C3×A4⋊S4 |
| Fitting | C2×C32×H(5,𝔽2) | C3⋊D3 |
| socle | C22×C3 | C3×A4⋊S4 |
| LCS | G ▷ C3×Ω+(4,𝔽3) | |
| UCS | C1 ◁ C22 | |
2221 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 2221 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(3,4,26,16,22,19,8,12,18,6,15,24)(5,25,20,14,10,21,13,11,17,7,23,9)(27,28,34)(29,31,33)(30,32,35);(4,9)(5,13)(6,14)(7,16)(10,20)(11,19)(12,21)(15,26)(17,23)(18,22)(24,25)(28,30)(29,32)(31,33)(34,35);(5,6,7)(10,11,15)(13,16,14)(17,18,24)(19,20,26)(22,23,25)142 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,26,16,22,19,8,12,18,6,15,24)(5,25,20,14,10,21,13,11,17,7,23,9)(27,28,34)(29,31,33)(30,32,35), (4,9)(5,13)(6,14)(7,16)(10,20)(11,19)(12,21)(15,26)(17,23)(18,22)(24,25)(28,30)(29,32)(31,33)(34,35), (5,6,7)(10,11,15)(13,16,14)(17,18,24)(19,20,26)(22,23,25));