Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2258 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×D3 |
| commutator, socle | C2×C3 | C22×C16 |
| Fitting | C3×#504 | C2 |
| prime radical | C3×#41 | C8 |
| cosocle | C3×C16 | C23 |
| hypercenter | #504 | D3 |
| hypocenter | C3 | #2258 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C16 ◁ #504 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3)(6,16)(11,26)(17,36)(18,37)(19,38)(27,46)(28,47)(29,48)(39,56)(40,57)(41,58)(49,61)(50,62)(51,63)(59,66)(64,67);(4,6)(5,11)(7,17)(8,18)(9,19)(10,16)(12,27)(13,28)(14,29)(15,26)(20,39)(21,40)(22,36)(23,41)(24,37)(25,38)(30,49)(31,50)(32,46)(33,51)(34,47)(35,48)(42,59)(43,56)(44,57)(45,58)(52,64)(53,61)(54,62)(55,63)(60,66)(65,67);(2,3)(4,5,7,12,8,13,20,30,9,14,21,31,23,33,42,52,10,15,22,32,24,34,43,53,25,35,44,54,45,55,60,65)(6,11,17,27,18,28,39,49,19,29,40,50,41,51,59,64,16,26,36,46,37,47,56,61,38,48,57,62,58,63,66,67)Polycyclic presentation:
2^7,3|a^2=d,d^2=e,e^2=f,f^2=g,h^a=h^2,c^b=cg120 conjugacy classes — view the full character table.
| maximal subgroup of | #10715748, #10715751, #10715753, … |
| maximal normal subgroup of | #10715748, #10715751, #10715753, … |
| maximal quotient of | #10715367, #10715384, #10715673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4027);