Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
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 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,5,7)(3,4,6,8)(9,20,21,17,13,14,18,15)(10,11,24,12,16,19,23,22);(2,3,5)(6,7,8)(11,18,12)(14,23,15)(17,20,24)(19,21,22)Polycyclic presentation:
2,3,2^6|a^2=h,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, #11711409, #11711413, … |
| maximal normal subgroup of | #11711408, #11711409, #11711413, … |
| maximal quotient of | #11714221, #11714229, #11714257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18125);