Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3390 |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(4,7,8,5)(6,9,11,10)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(12,14,17,22)(13,18,20,26)(15,23,16,25)(19,27,24,21);(4,6,8,11)(5,9,7,10)(14,22)(18,26)(21,27)(23,25);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,19)(16,24)(17,20)(18,23)(22,27)(25,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=de,f^c=fg,h^d=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3390, C2×#79820, #11675752, … |
| maximal normal subgroup of | D3×#3390, C2×#79820, #11675752, … |
| maximal quotient of | #10811874, #10811878, #10813508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12396);