Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #3352 |
336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 336 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)(6,11)(9,10)(13,19)(17,25)(18,20)(24,26);(4,7,8,5)(6,10,11,9)(13,18)(14,21)(16,23)(17,24)(19,20)(25,26);(2,3)(4,6)(5,9)(7,10)(8,11)(12,14)(13,17)(15,21)(16,22)(18,26)(19,25)(20,24)(23,27);(2,3)(4,5,8,7)(6,10,11,9)(12,13,15,19)(14,17,21,25)(16,18,23,20)(22,26,27,24)Polycyclic presentation:
2^7,3|a^2=e,c^2=eg,f^2=g,b^a=beg,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,d^b=deg,h^b=h^2,f^c=fg,h^d=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10708841, #10708845, #10708855, … |
| maximal normal subgroup of | #10708841, #10708845, #10708855, … |
| maximal quotient of | #10811600, #10812958, #10813413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12320);