Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C4⋊S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C4⋊S4 |
| commutator, cosocle | #41⋊C3 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | #131 | D3 |
| prime radical | #986 | C2 |
| hypercenter | C8 | S4 |
| hypocenter | SL(2,𝔽3) | D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ 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:
(1,8,27,7,16,24,11,31,15,2,12,30,6,23,28,20,4,22,3,18,17,10,25,32)(5,9,13,19,14,21,26,29);(2,7)(3,11)(4,5)(8,19)(9,18)(10,20)(12,26)(13,25)(14,16)(15,17)(21,31)(22,32)(23,29)(24,30)Polycyclic presentation:
2^3,3,2^3|b^2=c,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 | C2×#5406, #85451, #85454, … |
| maximal normal subgroup of | C2×#5406, #85451, #85454, … |
| maximal quotient of | #73012, #73016, #73021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 965);