Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3550 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| prime radical | #5548 | C2 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | #3550 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(12,16,17,15)(13,19,20,25)(14,22,23,24)(18,27,21,26);(2,3)(5,9)(8,10)(12,15,17,16)(13,19,20,25)(14,22,23,24)(18,26,21,27);(4,6,7,11)(5,8,9,10);(6,11)(8,10)(12,14,17,23)(13,18,20,21)(15,24,16,22)(19,27,25,26);(4,5)(6,10)(7,9)(8,11)(12,13,17,20)(14,21,23,18)(15,19,16,25)(22,27,24,26)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #10773117, #10773123, #10773164, … |
| maximal normal subgroup of | #10773117, #10773123, #10773164, … |
| maximal quotient of | #10958040, #10958091, #10958096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19331);