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 ⋊ #3561 |
| 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 | #3561 |
| 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)(12,15,16,24)(13,18,19,23)(14,22,21,27)(17,25,20,26);(2,3)(4,6,7,10)(5,9,8,11)(14,21)(17,20)(22,27)(25,26);(4,5)(6,9)(7,8)(10,11);(5,8)(9,11)(12,14)(13,17)(15,22)(16,21)(18,25)(19,20)(23,26)(24,27);(2,3)(5,8)(6,10)(12,13,16,19)(14,20,21,17)(15,23,24,18)(22,25,27,26)Polycyclic presentation:
2^7,3|a^2=f,d^2=g,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^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11688710, #11688714, #11688743, … |
| maximal normal subgroup of | #11688710, #11688714, #11688743, … |
| maximal quotient of | #10958129, #10958192, #10958434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19333);