Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#435 ⋊ C2 |
| Sylow | C3 ⋊ #3347 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#435 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #435 | D3 |
| hypocenter | C3 | #3347 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #435 | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,11)(9,10)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(2,3)(4,7,8,5)(6,10,11,9)(12,14)(13,18)(15,23)(16,25)(17,22)(19,27)(20,26)(21,24);(4,6)(5,9)(7,10)(8,11)(14,22)(18,26)(21,27)(23,25);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,19)(16,24)(17,20)(18,23)(22,27)(25,26)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=de,f^c=fg,h^c=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3347, C2×#79473, #11673847, … |
| maximal normal subgroup of | D3×#3347, C2×#79473, #11673847, … |
| maximal quotient of | #10811492, #10811582, #10812872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12049);