Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×(#1020⋊C3)⋊C2 (order 1152) · Inn(G) = C22⋊A4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #585 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3 |
| Frattini, socle | C2 | C2×C22⋊A4 |
| Fitting | #585 | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,8)(3,7,15,13,4,12,10,14,9,6,11,16);(3,10)(4,11)(6,13)(7,14);(1,9)(2,12)(3,4)(5,15)(6,7)(8,16)(10,11)(13,14)Polycyclic presentation:
2,3,2^5|a^2=g,c^b=d,d^b=cd,e^b=f,f^b=ef,f^c=fg,e^d=eg22 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1509);