Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C4×SD16 ⋊ C2 |
| Sylow | C3 ⋊ #3284 |
494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 494 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:
(1,2,3)(8,9)(10,19)(11,15)(12,21)(13,16)(14,22)(17,20)(18,23);(2,3)(5,7)(8,11,13,12)(9,15,16,21)(10,18,20,22)(14,19,23,17);(8,10,13,20)(9,14,16,23)(11,18,12,22)(15,19,21,17);(2,3)(4,5)(6,7)(8,9,13,16)(10,17,20,19)(11,15,12,21)(14,22,23,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=h^2,d^b=de,e^b=eg,h^c=h^2,e^d=eg63 conjugacy classes — view the full character table.
| maximal subgroup of | #10784588, #10784596, #10784603, … |
| maximal normal subgroup of | #10784588, #10784596, #10784603, … |
| maximal quotient of | #10785979, #10785994, #10786009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12251);