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 ⋊ #3541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | #3541 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
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)(8,11)(12,15,17,16)(13,19,20,25)(14,22,23,24)(18,26,21,27);(5,9)(6,10);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14)(13,18)(15,24)(16,22)(17,23)(19,27)(20,21)(25,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,23)(22,27)(24,26)Polycyclic presentation:
2^7,3|d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^b=h^2,h^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11680752, #11680756, #11680814, … |
| maximal normal subgroup of | #11680752, #11680756, #11680814, … |
| maximal quotient of | #10957985, #10958051, #10958508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19330);