Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C22×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | 2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23 |
| Fitting | 2−1+4 | D3 |
| hypercenter | D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ D4 | |
152 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,3,10)(2,13,15,6,7,16)(5,11,12)(8,14);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16);(3,4)(7,8)(9,11)(10,12)(13,14)(15,16)Polycyclic presentation:
2^3,3,2^3|e^2=g,f^2=g,d^a=d^2,e^a=f,f^a=e,c^b=cg,e^d=fg,f^d=ef,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #85469, #85470, #85473, … |
| maximal normal subgroup of | #85469, #85470, #85473, … |
| maximal quotient of | #85351, #85353, #85354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1485);