Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #520 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | #520 | D3 |
| prime radical | #5427 | C2 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | SL(2,𝔽3) | #41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
200 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 32 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)(9,24,26)(10,17,32,16,25,31)(11,20,38,19,33,36)(12,23,14)(15,37,40)(18,22)(21,34)(28,39,30);(2,5)(6,8)(9,11,14,19)(10,15,16,30)(12,20,24,33)(13,21,27,34)(17,28,25,37)(18,29,22,35)(23,36,26,38)(31,39,32,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,25)(14,16)(17,27)(18,24)(19,30)(20,35)(21,37)(23,32)(26,31)(28,34)(29,33)(36,40)(38,39)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85371, #11714815, #11714821, … |
| maximal normal subgroup of | C2×#85371, #11714815, #11714821, … |
| maximal quotient of | #11712667, #11712781, #11712784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17947);