Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×#5945 |
| Fitting, Sylow | C2×#585 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C2×#585 | C3 |
| socle | C22 | C23×A4 |
| hypercenter | C2×#41 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
1241 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1241 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 4 generators on 34 points:
(1,2)(3,5)(4,9)(6,15,16,14,7,29)(8,13)(10,22,23,21,11,33)(12,20)(17,28,18,27,19,30)(24,32,25,31,26,34);(3,23)(4,18)(5,33)(6,26)(7,10)(8,25)(9,30)(11,17)(12,16)(13,34)(14,32)(15,21)(19,24)(20,29)(22,27)(28,31);(5,13)(9,20)(14,27)(15,28)(21,31)(22,32)(29,30)(33,34);(1,2)76 conjugacy classes — view the full character table.
| maximal subgroup of | #60065⋊C3, #60070⋊C3, #60076⋊C3, … |
| maximal normal subgroup of | #60065⋊C3, #60070⋊C3, #60076⋊C3, … |
| maximal quotient of | C2×#3428⋊C3, C2×C4×#994, C2×C4×2−1+4⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20111);