Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #3278 |
482 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(8,11,13,9)(10,16,17,23)(12,15,22,20)(14,21,19,18);(5,7)(10,17)(14,19)(16,23)(18,21);(8,10,13,17)(9,14,11,19)(12,21,22,18)(15,23,20,16);(2,3)(4,5)(6,7)(8,9,13,11)(10,16,17,23)(12,20,22,15)(14,21,19,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,c^b=cg,d^b=de,e^b=eg,e^d=eg,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10784620, #10784622, #10784635, … |
| maximal normal subgroup of | #10784620, #10784622, #10784635, … |
| maximal quotient of | #10786520, #10786637, #10786766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16160);