Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×C4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3480 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4×Q8 | C2 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #3480 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(13,16)(17,19);(4,7,8,6)(5,9,10,11);(2,3)(4,6,8,7)(5,9,10,11)(12,13)(14,17)(15,16)(18,19);(4,5,8,10)(6,9,7,11)(13,16)(17,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^b=df,h^b=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | #10687214, #10687218, #10762069, … |
| maximal normal subgroup of | #10687214, #10687218, #10762069, … |
| maximal quotient of | #10954573, Q8×#926, #10954620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18378);