Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#460 ⋊ C2 |
| Sylow | C3 ⋊ #3216 |
390 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 27 points:
(1,2)(5,9)(8,10)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(6,11)(8,10)(12,14)(13,18)(15,23)(16,25)(17,22)(19,27)(20,26)(21,24);(2,3)(4,6,7,11)(5,8,9,10)(14,22)(18,26)(21,27)(23,25);(4,5)(6,10)(7,9)(8,11)(12,13,17,20)(14,21,22,27)(15,19,16,24)(18,25,26,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,h^b=h^2,f^c=fg,h^d=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10772062, #10772065, #10772084, … |
| maximal normal subgroup of | #10772062, #10772065, #10772084, … |
| maximal quotient of | #10804531, #10804946, #10805757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12965);