Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×D24 |
| Fitting | C3×C8×Q8 ⋊ C2 |
| Sylow | C3 ⋊ Q8×D8 |
404 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,13,17,19,15,18,14,16);(2,3)(13,16)(14,17)(18,19);(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)75 conjugacy classes — view the full character table.
| maximal subgroup of | #10708861, #10708865, #10708889, … |
| maximal normal subgroup of | #10708861, #10708865, #10708889, … |
| maximal quotient of | Q8×#821, C4⋊C4×D24, Q8×#818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12504);