Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3007 |
754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,11,13,12)(9,15,17,16)(10,19,20,21)(14,18,22,23);(5,7);(2,3)(8,10)(9,14)(11,19)(12,21)(13,20)(15,18)(16,23)(17,22);(4,5)(6,7)(8,9,13,17)(10,18,20,23)(11,16,12,15)(14,21,22,19)Polycyclic presentation:
2^7,3|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10692233, #10692239, #10692345, … |
| maximal normal subgroup of | #10692233, #10692239, #10692345, … |
| maximal quotient of | #10767588, #10767627, #10767666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15311);