Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #213 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#4515 |
| Fitting | C2×C3×M32 ⋊ C2 |
| Sylow | C3 ⋊ C2×#350 |
224 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 93 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 — 3 generators on 37 points:
(3,4,5)(6,8,9,18,10,20,22,33,11,21,23,34,24,19,35,32)(7,12,13,25,15,17,28,31,16,26,29,36,14,30,27,37);(4,5)(6,7)(8,17)(9,13)(10,15)(11,16)(12,19)(14,24)(18,31)(20,26)(21,30)(22,28)(23,29)(25,32)(27,35)(33,36)(34,37);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10710234, C4×#4515, #10710239, … |
| maximal normal subgroup of | #10710234, C4×#4515, #10710239, … |
| maximal quotient of | C2×#67523, #10704263, C2×#67525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4012);