Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4 |
| Fitting | C2×#41 | D3 |
| socle | C2 | C2×SL(2,ℤ/4) |
| prime radical | C2×#41⋊C3 | C2 |
| cosocle | #41⋊C3 | C22 |
| hypercenter | C2×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,4,15)(2,19,23,7,9,24)(3,11,26)(6,16,17)(8,29,31,18,21,32)(10,20)(13,27,28)(22,30);(1,2,3,8,6,7,13,18)(4,10,11,22,16,20,27,30)(5,9,12,21,14,19,25,29)(15,24,26,32,17,23,28,31)Polycyclic presentation:
2^3,3,2^3|a^2=c,c^2=g,e^2=g,f^2=g,b^a=bg,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #73073, #73120, #73129, … |
| maximal normal subgroup of | #73073, #73120, #73129, … |
| maximal quotient of | #73055, #73056, #73067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 982);