Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #2452 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2452 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(8,10,11,9)(12,14,15,19)(13,17,16,18);(2,3)(5,7)(8,9,11,10);(2,3)(4,5)(6,7)(8,9,11,10)(13,16)(17,18);(9,10)(12,13,15,16)(14,18,19,17)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=f,d^2=ef,b^a=bef,c^a=cf,d^a=def,c^b=cg,h^b=h^2,h^c=h^2,h^d=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2452, C2×#82766, #10963105, … |
| maximal normal subgroup of | D3×#2452, C2×#82766, #10963105, … |
| maximal quotient of | #10717209, #10717557, #10718067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15342);