Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/8) (order 768) · Inn(G) = C4⋊S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C4⋊S4 |
| commutator, prime radical, cosocle | #41⋊C3 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | #131 | D3 |
| hypercenter | C8 | S4 |
| hypocenter | SL(2,𝔽3) | D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,16,50,26,24,46,4,41,23,3,39,64,7,45,49,15,5,44,14,51,25,18,20,63)(2,27,58,38,34,56,9,36,33,8,47,62,12,55,57,13,10,54,19,61,35,29,30,60)(6,17,40,52,22,43,21,42)(11,28,48,59,32,53,31,37);(1,2,7,12)(3,13,18,38)(4,19,14,9)(5,11,24,32)(6,10,22,34)(8,26,29,15)(16,37,45,59)(17,36,43,61)(20,48,39,31)(21,47,40,30)(23,35,25,33)(27,52,55,42)(28,51,53,41)(44,62,46,60)(49,57,50,58)(54,63,56,64)Polycyclic presentation:
2^3,3,2^3|a^2=g,b^2=cg,c^2=g,e^2=g,f^2=g,b^a=bc,c^a=cg,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg20 conjugacy classes — view the full character table.
| maximal subgroup of | #73041, #73042, #73138, … |
| maximal normal subgroup of | #73041, #73042, #73138, … |
| maximal quotient of | #73011, #73013, #73015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 962);