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 ⋊ #3552 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | #3552 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊Q8 | |
1004 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)(5,8)(9,11);(4,6,7,10)(5,9,8,11)(13,19)(17,26)(18,22)(25,27);(12,15,16,24)(13,18,19,22)(14,20,21,23)(17,25,26,27);(5,8)(6,10)(12,14,16,21)(13,17,19,26)(15,23,24,20)(18,27,22,25);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,22)(16,19)(18,24)(20,27)(21,26)(23,25)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df66 conjugacy classes — view the full character table.
| maximal subgroup of | #10687584, #10687588, #10773089, … |
| maximal normal subgroup of | #10687584, #10687588, #10773089, … |
| maximal quotient of | #10958070, #10958231, #10958760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18794);