Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×#216 |
| Sylow | C3 ⋊ C2×C4×Q16 |
488 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(4,5)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19);(1,2)(6,7,8,9);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10689752, #10689753, #10689755, … |
| maximal normal subgroup of | #10689752, #10689753, #10689755, … |
| maximal quotient of | C2×C8×#216, C2×#70956, C2×#70934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15001);