Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#4768 |
| Sylow | C3 ⋊ C2×#488 |
412 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,4,5)(6,9,10,8)(7,12,13,11)(15,20)(16,23)(17,26)(19,27)(22,29)(24,25);(1,2)(6,8)(7,11)(9,10)(12,13)(14,16)(15,19)(17,23)(18,25)(20,22)(21,29)(24,26)(27,28);(4,5)(6,7,10,13)(8,12,9,11)(14,15)(16,22)(17,21)(18,20)(19,25)(23,27)(24,29)(26,28);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4768, #10803175, #10803179, … |
| maximal normal subgroup of | C4×#4768, #10803175, #10803179, … |
| maximal quotient of | C2×#68848, C2×#68850, C2×#68851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11310);