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 ⋊ #3333 |
398 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 398 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)(6,11)(8,10)(12,15,16,17)(13,19,20,21)(14,24,23,25)(18,22,26,27);(12,14,16,23)(13,18,20,26)(15,24,17,25)(19,22,21,27);(2,3)(4,6)(5,8)(7,11)(9,10)(14,23)(18,26)(22,27)(24,25);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,20)(14,22,23,27)(15,21,17,19)(18,25,26,24)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=g,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 | #10784924, #10784926, #10784984, … |
| maximal normal subgroup of | #10784924, #10784926, #10784984, … |
| maximal quotient of | #10810852, #10811638, #10812433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12316);