Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #1020 |
| commutator, hypocenter | #997 | C4 |
| Fitting | #585 | D3 |
| socle | C2 | #5936 |
| prime radical, cosocle | #5950 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C4 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,8,23,29,6,21,31,18)(2,11,27,5,10,24,32,16)(3,17,26,9,13,28,14,22)(4,20,25,7,15,30,12,19);(4,5,14)(8,9,20)(11,12,23)(15,16,26)(17,18,27)(21,22,30)(24,25,31)(28,29,32)Polycyclic presentation:
2^2,3,2^5|a^2=b,b^2=h,c^a=c^2,d^a=e,e^a=d,f^a=g,g^a=f,d^c=e,e^c=de,f^c=fg,g^c=f,f^d=fh,g^e=gh26 conjugacy classes — view the full character table.
| maximal subgroup of | #11711588, #11711590, #11711596, … |
| maximal normal subgroup of | #11711588, #11711590, #11711596, … |
| maximal quotient of | #11714225, #11714228, #11714230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18142);