Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | #585 | D3 |
| socle | C2 | C22×SL(2,ℤ/4) |
| prime radical | #5945 | C2 |
| cosocle | #41⋊C3 | C23 |
| hypercenter | #41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ #41 | |
304 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,17,40)(2,21,39)(3,34,48,13,31,46)(4,22,18)(5,38,11)(6,29,37)(7,20,36)(8,42,47,23,35,43)(9,10,19)(12,27,28)(14,15,32,26,30,44)(16,25,45,24,41,33);(1,3)(2,8)(4,14)(5,15)(6,16)(7,13)(9,24)(10,25)(11,26)(12,23)(17,31)(18,32)(19,33)(20,34)(21,35)(22,30)(27,42)(28,43)(29,41)(36,46)(37,45)(38,44)(39,47)(40,48);(1,2,4,9,7,12,11,6)(3,8,14,24,13,23,26,16)(5,18,17,36,22,38,20,40)(10,28,27,37,29,39,21,19)(15,32,31,46,30,44,34,48)(25,43,42,45,41,47,35,33)Polycyclic presentation:
2^4,3,2^3|a^2=d,d^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #11712953, #11712955, #11713094, … |
| maximal normal subgroup of | #11712953, #11712955, #11713094, … |
| maximal quotient of | #11711788, #11711789, #11711800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18117);