Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Sylow | #2811 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2811 | C3 |
| prime radical | C22⋊SL(2,𝔽3) | C22 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,11,22,6,4,19,7,15,12,3,13,23)(2,14,20,9,8,21,10,18,16,5,17,24);(4,11,13,15)(8,14,17,18)(12,21,22,24)(16,23,20,19)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=h,e^2=h,b^a=bh,d^a=dfh,e^a=egh,d^b=dgh,e^b=efg,d^c=defh,e^c=dfg,f^c=fgh,g^c=f,e^d=eh24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73203, #11710115, #52914⋊C3, … |
| maximal normal subgroup of | C2×#73203, #11710115, #52914⋊C3, … |
| maximal quotient of | #14601⋊C3, #17218⋊C3, #11709342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5779);