Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3276 |
472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 472 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)(8,12)(9,11)(10,15)(13,17)(14,16)(18,19)(20,23)(21,22);(5,7)(8,11,14,13)(9,16,17,12)(10,19,21,20)(15,18,22,23);(8,10,14,21)(9,15,17,22)(11,20,13,19)(12,18,16,23);(2,3)(4,5)(6,7)(8,9,14,17)(10,18,21,23)(11,16,13,12)(15,19,22,20)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^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 | #10784565, #10784567, #10784610, … |
| maximal normal subgroup of | #10784565, #10784567, #10784610, … |
| maximal quotient of | #10786500, #10786607, #10786714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16136);