Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #2015 |
390 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(9,11)(13,19)(15,16)(18,26)(20,21)(22,27)(23,24);(1,2,3)(5,7)(8,9)(10,11)(12,14,17,25)(13,18,21,27)(15,24,16,23)(19,26,20,22);(4,5)(6,7)(12,13,17,21)(14,22,25,26)(15,20,16,19)(18,24,27,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,e^b=eg,d^c=dg,e^c=eg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2015, C2×#72072, #10822254, … |
| maximal normal subgroup of | D3×#2015, C2×#72072, #10822254, … |
| maximal quotient of | #10695668, #10695674, #10695688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4648);