Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×S4 (order 192) · Inn(G) = C2×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4 |
| Fitting | #131 | D3 |
| socle | C2 | C4×S4 |
| prime radical | #985 | C2 |
| cosocle | #41⋊C3 | C22 |
| hypercenter | C8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ C8 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,9,26,8,16,24,13,29,15,2,11,31,6,19,28,18,4,23,3,21,17,7,27,32)(5,10,12,22,14,20,25,30);(2,7)(4,5)(8,18)(9,20)(10,19)(11,12)(14,16)(15,17)(21,30)(22,29)(25,27)(26,28)Polycyclic presentation:
2^3,3,2^3|b^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 | C2×#5405, #85448, #85452, … |
| maximal normal subgroup of | C2×#5405, #85448, #85452, … |
| maximal quotient of | #226⋊C8, #73010, #73018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 964);