Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×SO+(4,𝔽3) |
| Fitting | C22×H(5,𝔽2) ⋊ C3⋊D3 |
| derived | Ω+(4,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C23 |
| Frattini | C2 | C22×A4⋊S4 |
| Fitting | C22×H(5,𝔽2) | C3⋊D3 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C23 | |
2364 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 2364 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 28 points:
(3,4)(5,9,18)(6,14,13)(7,17,11)(16,27,20)(21,28,26)(22,25,24);(1,2)(5,6,12)(7,15,22)(8,13,25)(9,19,28)(10,20,26)(11,16,23)(14,21,17)(18,27,24);(6,12)(7,9)(8,18)(10,17)(13,24)(14,26)(15,28)(16,23)(19,22)(20,21)(25,27)80 conjugacy classes — view the full character table.
| maximal subgroup of | C23×SO+(4,𝔽3), #72226130, #72226133, … |
| maximal normal subgroup of | C23×SO+(4,𝔽3), #72226130, #72226133, … |
| maximal quotient of | C23×SO+(4,𝔽3), D4×SO+(4,𝔽3), C2×D3×SO+(4,𝔽3), … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,9,18)(6,14,13)(7,17,11)(16,27,20)(21,28,26)(22,25,24), (1,2)(5,6,12)(7,15,22)(8,13,25)(9,19,28)(10,20,26)(11,16,23)(14,21,17)(18,27,24), (6,12)(7,9)(8,18)(10,17)(13,24)(14,26)(15,28)(16,23)(19,22)(20,21)(25,27));