Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#5194 |
| Fitting | C2×C3×#135 ⋊ C2 |
| Sylow | C3 ⋊ C2×#508 |
368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(3,4,5)(7,13)(9,20)(10,21)(12,24)(14,15)(16,29)(17,30)(18,31)(22,34)(23,28)(25,26)(27,36)(32,37)(33,35);(1,2)(6,8,11,19)(7,12,16,27)(9,17,22,32)(10,18,21,31)(13,24,29,36)(14,25,28,35)(15,26,23,33)(20,30,34,37);(4,5)(6,7)(8,12)(9,13)(10,23)(11,16)(14,20)(15,21)(17,24)(18,33)(19,27)(22,29)(25,30)(26,31)(28,34)(32,36)(35,37);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10712234, #10712235, #10712237, … |
| maximal normal subgroup of | #10712234, #10712235, #10712237, … |
| maximal quotient of | C2×#70651, C2×#72077, C2×#70648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16468);