Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×C4×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #2357 |
560 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,10,11,14)(9,13,12,15);(9,12)(13,15)(16,19,20,18)(17,21,22,23);(2,3)(9,12)(10,14)(16,18,20,19)(17,21,22,23);(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,23)(20,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=ef,c^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de,h^b=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | #10688782, #10688787, #10691107, … |
| maximal normal subgroup of | #10688782, #10688787, #10691107, … |
| maximal quotient of | #10716631, #10716650, #10716675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6221);