Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C4×SO+(4,𝔽3) |
| Fitting | C4×H(5,𝔽2) ⋊ C3⋊D3 |
| derived | Ω+(4,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | A4⋊S4 |
| commutator, hypocenter | Ω+(4,𝔽3) | C2×C4 |
| Frattini, socle | C22 | C2×A4⋊S4 |
| Fitting | C4×H(5,𝔽2) | C3⋊D3 |
| prime radical | C2×SO+(4,𝔽3) | C2 |
| cosocle | C2×Ω+(4,𝔽3) | C22 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2×C4 | |
836 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 43 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:
(5,11)(6,24,20,28)(7,17,10,8)(9,18)(12,15,21,13)(14,27,16,22)(19,23,25,26);(1,2,3,4)(5,6,12)(7,14,27)(8,15,21)(9,19,24)(10,20,28)(11,16,23)(13,22,18)(17,25,26);(7,8,18)(9,10,17)(13,14,15)(19,20,25)(21,22,27)(24,28,26)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226133, D4×SO+(4,𝔽3), #72226335, … |
| maximal normal subgroup of | #72226133, D4×SO+(4,𝔽3), #72226335, … |
| maximal quotient of | AGL(1,𝔽5)×SO+(4,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,11)(6,24,20,28)(7,17,10,8)(9,18)(12,15,21,13)(14,27,16,22)(19,23,25,26), (1,2,3,4)(5,6,12)(7,14,27)(8,15,21)(9,19,24)(10,20,28)(11,16,23)(13,22,18)(17,25,26), (7,8,18)(9,10,17)(13,14,15)(19,20,25)(21,22,27)(24,28,26));