Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #2650 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2650 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
386 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)(12,14,15,18)(13,17,16,19)(20,23,24,22)(21,25,26,27);(2,3)(6,10)(8,11)(13,16)(17,19)(21,26)(22,23);(4,6)(5,8)(7,10)(9,11)(20,22,24,23)(21,25,26,27);(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,24,26)(22,27,23,25)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2650, C2×#73814, #10961880, … |
| maximal normal subgroup of | D3×#2650, C2×#73814, #10961880, … |
| maximal quotient of | #10721656, #10721686, #10721754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6390);