Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = C2×C4×S4 (order 192) · Inn(G) = S4 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C8 |
| Fitting | #131 | D3 |
| socle | C2 | A4⋊C8 |
| prime radical | #41⋊C3 | C4 |
| cosocle | #131⋊C3 | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,10,41,26,19,55,58,61,7,33,39,49,4,28,60,47)(2,14,52,18,30,59,62,6,12,40,50,42,9,35,64,20)(3,25,45,29,37,63,23,11,16,51,43,53,13,46,21,31)(5,34,15,27,44,54,57,24,22,32,38,8,17,56,36,48);(5,20,23)(6,21,22)(10,31,34)(11,32,33)(14,38,41)(15,39,40)(17,42,45)(18,43,44)(25,49,52)(26,50,51)(28,53,56)(29,54,55)(35,57,60)(36,58,59)(46,61,64)(47,62,63)Polycyclic presentation:
2^3,3,2^3|a^2=b,b^2=c,c^2=g,e^2=g,f^2=g,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #73038, #73039, C2×#4628, … |
| maximal normal subgroup of | #73038, #73039, C2×#4628, … |
| maximal quotient of | SL(2,𝔽3)⋊C16, C2×#4628, #10588308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 187);