Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#5065 |
| Sylow | C3 ⋊ C2×#499 |
360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,4,5)(6,8,9,13)(7,10,11,12)(14,16)(15,19)(17,25)(18,23)(20,28)(21,27)(22,29)(24,26);(6,7,9,11)(8,12,13,10)(16,23)(19,27)(22,28)(24,25);(4,5)(14,15,17,20,18,21,26,29)(16,22,24,27,23,28,25,19);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10694732, #10694734, #10694756, … |
| maximal normal subgroup of | #10694732, #10694734, #10694756, … |
| maximal quotient of | C2×#69549, C2×#69618, C2×#69619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15589);