Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = C23×S4 (order 192) · Inn(G) = S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4 |
| Frattini | C4 | C2×S4 |
| Fitting | #131 | D3 |
| socle | C2 | C4×S4 |
| prime radical | #985 | C2 |
| cosocle | #41⋊C3 | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,8,26,7,16,24,13,31,15,2,11,30,6,23,28,20,4,22,3,18,17,10,27,32)(5,9,12,19,14,21,25,29);(4,5)(8,9)(11,12)(14,16)(15,17)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^3,3,2^3|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×#5404, … |
| maximal normal subgroup of | #73038, #73039, C2×#5404, … |
| maximal quotient of | C8×#226, #73009, C8×GL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 963);