Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3524⋊C3 |
| Fitting, Sylow | #3524 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3524 | C3 |
| hypercenter | #126 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #126 | |
255 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,3,4,7)(2,6,5,8)(10,16)(13,24,25)(14,26,23)(15,19)(17,18,31,29,30,32)(21,36,37)(22,38,35)(27,28,39,33,34,40);(2,5)(6,8)(9,38)(10,40)(11,25)(12,37)(13,22)(14,35)(15,31)(16,39)(17,28)(18,33)(19,32)(20,26)(21,24)(23,36)(27,30)(29,34);(1,2)(3,6)(4,5)(7,8)(9,10,12,16)(11,19,20,15)(13,17,23,29)(14,18,24,30)(21,33,35,27)(22,34,36,28)(25,31,26,32)(37,40,38,39)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=h,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=gh,g^d=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3524⋊C3, #62793⋊C3, #62810⋊C3, … |
| maximal normal subgroup of | C2×#3524⋊C3, #62793⋊C3, #62810⋊C3, … |
| maximal quotient of | #34197⋊C3, #34262⋊C3, #34263⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18197);