Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#520 ⋊ C2 |
| Sylow | C3 ⋊ C4×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting | C3×#520 | C2 |
| socle, cosocle | C22×C3 | C25 |
| prime radical | C2×#1010 | C2 |
| hypercenter | #520 | D3 |
| hypocenter | C3 | C4×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #520 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(4,6,7,5);(9,15)(13,16)(14,23)(21,22);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22);(2,3)(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21);(4,5,7,6)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=f,e^2=g,b^a=bf,d^a=df,c^b=cf,h^b=h^2,h^e=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #10801080, #10801086, #10801091, … |
| maximal normal subgroup of | #10801080, #10801086, #10801091, … |
| maximal quotient of | #10953425, #10953428, Dic3×#520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18463);