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 ⋊ #3396 |
384 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 384 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:
(1,2,3)(4,7,8,6)(5,10,9,11);(4,6,8,7)(5,10,9,11)(12,15,17,23)(13,19,20,27)(14,21,16,25)(18,26,24,22);(2,3)(4,5,8,9)(6,10,7,11)(12,14,17,16)(13,18,20,24)(15,21,23,25)(19,26,27,22);(5,9)(10,11)(12,13,17,20)(14,18,16,24)(15,22,23,26)(19,25,27,21)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,c^2=eg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,h^b=h^2,f^c=fg63 conjugacy classes — view the full character table.
| maximal subgroup of | #10708701, #10708705, #10708729, … |
| maximal normal subgroup of | #10708701, #10708705, #10708729, … |
| maximal quotient of | #10811999, #10814031, #10814941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12288);