Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#5431 |
| Fitting | C2×2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×2−1+4 | D3 |
| hypercenter | C2×D4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D4 | |
311 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(3,6,18)(4,21,26,9,11,33)(5,29,31,14,15,32)(8,19,20)(10,27,34,13,23,28)(12,22)(16,30)(24,25);(3,4)(5,9)(6,12)(7,11)(8,13)(10,14)(15,22)(16,21)(17,27)(18,28)(19,25)(20,26)(23,30)(24,29)(31,34)(32,33);(1,2)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5431, #11712834, #11713089, … |
| maximal normal subgroup of | C4×#5431, #11712834, #11713089, … |
| maximal quotient of | C2×#73088, C2×#73086, C2×#73096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18108);