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×#4561 |
| Sylow | C3 ⋊ C2×#350 |
224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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)(7,12)(9,20)(13,23)(14,16)(15,24)(17,28)(21,33)(25,27)(26,34)(29,30)(31,37)(35,36);(4,5)(6,7,8,13,10,15,18,26,11,16,19,27,22,30,32,36)(9,12,17,23,21,24,31,34,20,14,28,25,33,29,37,35);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4561, Dic3×#350, #10710240, … |
| maximal normal subgroup of | C4×#4561, Dic3×#350, #10710240, … |
| maximal quotient of | C2×#67689, #10704262, C2×#67718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4581);