Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #3139 |
380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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)(14,24)(15,25)(19,26)(20,27);(5,9)(8,11)(12,15,16,24,18,25,17,14)(13,20,21,26,23,27,22,19);(4,6,7,10)(5,8,9,11)(12,14,17,25,18,24,16,15)(13,19,22,27,23,26,21,20);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,18,23)(14,20,24,27)(15,19,25,26)(16,22,17,21)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,d^b=dg,d^c=dg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10692708, #10692728, #10692790, … |
| maximal normal subgroup of | #10692708, #10692728, #10692790, … |
| maximal quotient of | #10795402, #10797904, #10798868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11113);