Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of D96 |
| automorphism | Aut(G) = D3×#12067 (order 1536) · Inn(G) = D48 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C32 ⋊ C2 |
| Sylow | C3 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D48 |
| commutator, cosocle | C3×C16 | C22 |
| Frattini | C16 | C2×D3 |
| Fitting | C3×C32 | C2 |
| socle | C2×C3 | D16 |
| prime radical | D48 | C2 |
| hypercenter | C32 | D3 |
| hypocenter | C3 | SD64 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5,11,23,20,28,32,29,8,13,26,35,19,34,6,10,9,14,15,31,7,12,17,24,21,30,33,27,16,22,18,25);(2,3)(5,10)(6,15)(7,19)(8,21)(11,18)(12,27)(13,29)(14,25)(16,20)(17,26)(22,31)(23,34)(24,30)(28,35)(32,33)Polycyclic presentation:
2^6,3|b^2=cd,c^2=de,d^2=e,e^2=f,b^a=bc,c^a=cdf,d^a=def,e^a=ef,g^a=g^251 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4449, #69369, #69373, … |
| maximal normal subgroup of | C2×#4449, #69369, #69373, … |
| maximal quotient of | #67592, #67596, #67597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 8);