Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #1020 |
| commutator, prime radical, cosocle, hypocenter | #543⋊C3 | C2 |
| Fitting | #543 | D3 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C22 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,6,3,4,7,8,5)(9,14,23,15)(10,18,24,19)(11,20,12,16)(13,21,17,22);(2,6,3)(5,7,8)(11,12,17)(14,15,20)(18,19,23)(21,22,24)Polycyclic presentation:
2,3,2^6|a^2=gh,e^2=h,f^2=h,b^a=b^2,c^a=d,d^a=c,e^a=fh,f^a=eh,c^b=d,d^b=cd,e^b=ef,f^b=e,e^c=eg,f^d=fg,f^e=fh21 conjugacy classes — view the full character table.
| maximal subgroup of | #11711408, #11711410, #11711415, … |
| maximal normal subgroup of | #11711408, #11711410, #11711415, … |
| maximal quotient of | #11714222, #11714223, #11714229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18126);