Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#4564 |
| Sylow | C3 ⋊ C2×#358 |
232 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 232 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) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 25 points:
(3,4,5)(10,11,14,17)(12,19,21,25)(13,22,20,16)(15,23,24,18);(4,5)(6,7,8,9)(11,15)(12,18)(13,20)(16,25)(17,24)(19,22)(21,23);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4564, Dic3×#358, #10710406, … |
| maximal normal subgroup of | C4×#4564, Dic3×#358, #10710406, … |
| maximal quotient of | C2×#67765, #10705166, #10705218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4636);