Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3584⋊C3 |
| Fitting, Sylow | #3584 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×A4 |
| commutator, cosocle | #41 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | #3584 | C3 |
| socle | C2 | C2×D4×A4 |
| hypercenter | #135 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
334 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,4,8,5)(2,10,13,21)(3,15,18,22)(6,24,26,23,25,48,27,20,28,19,7,47)(9,29,32,14)(11,50,36,49,35,60,37,34,38,33,12,59)(16,52,44,51,43,64,45,42,46,41,17,63)(30,40,56,39,55,61,57,54,58,53,31,62);(1,46)(2,58)(3,26)(4,64)(5,63)(6,17)(7,45)(8,44)(9,36)(10,61)(11,31)(12,57)(13,56)(14,60)(15,47)(16,25)(18,28)(19,42)(20,51)(21,62)(22,48)(23,52)(24,41)(27,43)(29,59)(30,35)(32,38)(33,54)(34,39)(37,55)(40,49)(50,53);(1,2,8,13)(3,14,18,29)(4,10,5,21)(6,11,27,37)(7,12,25,35)(9,22,32,15)(16,39,45,53)(17,40,43,54)(19,33,23,49)(20,34,24,50)(26,36,28,38)(30,51,57,41)(31,52,55,42)(44,61,46,62)(47,59,48,60)(56,64,58,63)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=h,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,e^a=eh,e^b=eh,f^d=gh,g^d=fg,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #33385⋊C3, #33388⋊C3, #60183⋊C3, … |
| maximal normal subgroup of | #33385⋊C3, #33388⋊C3, #60183⋊C3, … |
| maximal quotient of | #60086⋊C3, #60108⋊C3, C4×#576⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18215);