Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2264 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#826 |
| commutator, cosocle | C3×C16 | C23 |
| Frattini | C16 | C22×D3 |
| Fitting, prime radical | C3×#508 | C2 |
| socle | C2×C3 | C2×D16 |
| hypercenter | #508 | D3 |
| hypocenter | C3 | #2264 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #508 | |
104 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)(5,12)(11,33)(13,36)(14,37)(15,31)(16,43)(21,42)(27,61)(34,63)(35,41)(38,64)(39,52)(40,65)(44,51)(48,67)(53,59);(2,3)(4,6,10,20)(5,11,12,33)(7,22,26,56)(8,28,30,62)(9,32,29,19)(13,38,36,64)(14,40,37,65)(15,41,31,35)(16,42,43,21)(17,45,47,24)(18,49,50,23)(25,60,58,55)(27,34,61,63)(39,67,52,48)(44,53,51,59)(46,57,54,66);(4,5)(6,16)(7,21)(8,27)(9,31)(10,12)(11,17)(13,24)(14,23)(15,29)(18,48)(19,51)(20,43)(22,40)(25,59)(26,42)(28,38)(30,61)(32,44)(33,47)(34,55)(35,54)(36,45)(37,49)(39,66)(41,46)(50,67)(52,57)(53,58)(56,65)(60,63)(62,64)Polycyclic presentation:
2^7,3|b^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,d^b=de,e^b=ef,f^b=fg,h^b=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10715802, #10715805, #10715865, … |
| maximal normal subgroup of | #10715802, #10715805, #10715865, … |
| maximal quotient of | #10715626, #10715633, #10715695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4086);