Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C22×D3×C16 (order 384) · Inn(G) = C2×D3 · Out(G) = C2×C16 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C64 ⋊ C2 |
| Sylow | C3 ⋊ M128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C2×D3 |
| commutator, socle | C2×C3 | C2×C32 |
| Fitting | C3×C64 | C2 |
| prime radical | C2×D3 | C32 |
| cosocle | C3×C32 | C22 |
| hypercenter | C64 | D3 |
| hypocenter | C3 | M128 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C32 ◁ C64 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(2,3)(5,11)(12,26)(13,27)(14,28)(15,29)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(52,61)(53,62)(54,63)(55,64)(65,67);(1,2,3)(4,5,6,12,7,13,16,30,8,14,17,31,20,33,36,52,9,15,18,32,21,34,37,53,23,35,39,54,42,55,56,65,10,11,19,26,22,27,38,46,24,28,40,47,43,49,57,61,25,29,41,48,44,50,58,62,45,51,59,63,60,64,66,67)Polycyclic presentation:
2^7,3|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,b^a=bg,h^b=h^2144 conjugacy classes — view the full character table.
| maximal subgroup of | D3×M128, C2×#67432, #10716225, … |
| maximal normal subgroup of | D3×M128, C2×#67432, #10716225, … |
| maximal quotient of | #10682991, (C2×D3)⋊C64, Dic3⋊C64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8);