Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3256 |
354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(2,3)(4,6,7,10)(5,9,8,11)(13,19)(17,21)(18,20)(25,26);(1,2,3)(5,8)(9,11)(13,18)(16,24)(17,25)(19,20)(21,26)(23,27);(4,5)(6,9)(7,8)(10,11)(12,14,15,22)(13,17,19,21)(16,23,24,27)(18,25,20,26);(5,8)(6,10)(12,13)(14,21)(15,19)(16,18)(17,22)(20,24)(23,26)(25,27)Polycyclic presentation:
2^7,3|b^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,f^c=fg,h^d=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | #10707721, #10707725, #10708571, … |
| maximal normal subgroup of | #10707721, #10707725, #10708571, … |
| maximal quotient of | #10805145, #10805591, #10807345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13432);