Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #550 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | #550 | D3 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | SL(2,𝔽3) | #41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
207 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 207 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)(11,35,37)(12,23,14)(15,29,40,28,36,39)(18,22)(19,34,21)(30,33);(2,5)(3,7)(9,11,14,21)(10,15,16,28)(12,19,24,35)(13,20,27,38)(17,29,25,36)(18,30,22,33)(23,34,26,37)(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)(19,33)(20,36)(21,28)(23,32)(26,31)(29,38)(30,35)(34,40)(37,39)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=cd,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85413, #11715018, #11715028, … |
| maximal normal subgroup of | C2×#85413, #11715018, #11715028, … |
| maximal quotient of | #11712909, #11712988, #11712993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17989);