Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4519 |
| automorphism | Aut(G) = D3×#12067 (order 1536) · Inn(G) = #826 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #826 |
| commutator, cosocle | C3×C16 | C22 |
| Frattini | C16 | C2×D3 |
| Fitting | C3×Q32 | C2 |
| socle, prime radical | C2×C3 | D16 |
| hypercenter | Q32 | D3 |
| hypocenter | C3 | Q64 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3)(4,6,10,20)(5,11,15,36)(7,22,26,56)(8,28,30,62)(9,32,29,19)(12,38,40,65)(13,42,43,67)(14,44,31,35)(16,45,47,21)(17,37,33,24)(18,41,50,23)(25,60,58,55)(27,34,61,63)(39,66,52,49)(46,53,51,59)(48,57,54,64);(2,3)(4,5,10,15)(6,16,20,47)(7,21,26,45)(8,27,30,61)(9,31,29,14)(11,33,36,17)(12,37,40,24)(13,41,43,23)(18,49,50,66)(19,51,32,46)(22,42,56,67)(25,59,58,53)(28,38,62,65)(34,60,63,55)(35,48,44,54)(39,57,52,64)Polycyclic presentation:
2^6,3|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,c^b=cd,d^b=de,e^b=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69379, #69381, D3×Q64, … |
| maximal normal subgroup of | #69379, #69381, D3×Q64, … |
| maximal quotient of | #67575, #67578, #67826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 81);