Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3393 |
360 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(1,2,3)(4,7,8,6)(5,10,9,11);(4,6,8,7)(5,10,9,11)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(2,3)(4,5,8,9)(6,10,7,11)(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(5,9)(10,11)(12,13)(14,17)(15,18)(16,20)(19,23)(21,24)(22,26)(25,27)Polycyclic presentation:
2^7,3|b^2=eg,c^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,h^b=h^2,f^c=fg63 conjugacy classes — view the full character table.
| maximal subgroup of | #10708715, #10708719, #10708743, … |
| maximal normal subgroup of | #10708715, #10708719, #10708743, … |
| maximal quotient of | #10811981, #10814031, #10814933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12155);