Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×#11921471 |
| derived | Ω+(4,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | A4⋊S4 |
| commutator, hypocenter | Ω+(4,𝔽3) | C2×C4 |
| Frattini | C22 | C2×A4⋊S4 |
| Fitting | C22×H(5,𝔽2) | C3⋊D3 |
| prime radical | C22×Ω+(4,𝔽3) | C2 |
| cosocle | C2×Ω+(4,𝔽3) | C22 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C23 | |
1074 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2)(7,12)(8,15,27,16,29,14)(9,10,11,19,17,20)(13,30,22,25,26,23)(18,28)(21,24);(3,4,5,6)(8,13)(9,17)(10,19)(11,20)(14,23)(15,24)(16,25)(18,30)(21,29)(22,27)(26,28);(7,8,13)(9,18,22)(10,14,26)(11,15,21)(12,16,25)(17,27,30)(19,28,23)(20,29,24)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72229366 |
| maximal quotient of | C2×#72229366 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(7,12)(8,15,27,16,29,14)(9,10,11,19,17,20)(13,30,22,25,26,23)(18,28)(21,24), (3,4,5,6)(8,13)(9,17)(10,19)(11,20)(14,23)(15,24)(16,25)(18,30)(21,29)(22,27)(26,28), (7,8,13)(9,18,22)(10,14,26)(11,15,21)(12,16,25)(17,27,30)(19,28,23)(20,29,24));