Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×#131⋊C3 |
| Fitting, Sylow | C22×#131 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C4 | C23×A4 |
| Fitting | C22×#131 | C3 |
| socle | C23 | C4×A4 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | #41 | C23×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C8 | |
359 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,7,11,10,14,17,24)(8,13,29,22,21,27,30,36,18,12,16,33,19,28,31,34,9,25,15,23,20,26,32,35);(3,4)(5,18,10,20)(6,25,14,27)(7,29,17,31)(8,21,19,9)(11,33,24,35)(12,28,26,13)(15,32,30,16)(22,36,34,23);(1,2)112 conjugacy classes — view the full character table.
| maximal subgroup of | #20336⋊C3, #20348⋊C3, #20351⋊C3, … |
| maximal normal subgroup of | #20336⋊C3, #20348⋊C3, #20351⋊C3, … |
| maximal quotient of | C2×C4×#131⋊C3, C2×#2880⋊C3, C2×C4⋊(#131⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18203);