Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | Aut(G) = C22×#1020 (order 384) · Inn(G) = #1020 · Out(G) = C22 |
| 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 | Q8⋊SL(2,𝔽3) | C2 |
| Fitting | Q82 | D3 |
| LCS | G ▷ Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,7,6,3,4,2,8,5)(9,20,21,17,13,14,18,15)(10,11,24,12,16,19,23,22);(2,6,3)(5,7,8)(11,12,21)(14,15,24)(17,23,20)(18,19,22)Polycyclic presentation:
2,3,2^6|a^2=h,c^2=h,d^2=h,e^2=g,f^2=g,b^a=b^2,c^a=dh,d^a=ch,e^a=fg,f^a=eg,c^b=d,d^b=cdh,e^b=ef,f^b=e,d^c=dh,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85552, #11715617, #11715618, … |
| maximal normal subgroup of | C2×#85552, #11715617, #11715618, … |
| maximal quotient of | #11714221, #11714222, #11714233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18128);