Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×S42 (order 1152) · Inn(G) = C22×S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23 |
| Fitting | H(5,𝔽2) | D3 |
| hypercenter | Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,12,18,14,4,28,6,29,16,3,17,30)(2,19,24,11,8,31,10,25,22,7,23,32)(5,13,15,27)(9,20,21,26);(1,2,6,10)(3,11,14,7)(4,8,17,23)(5,9,15,21)(12,25,29,19)(13,26,27,20)(16,22,18,24)(28,32,30,31);(4,5)(8,9)(12,13)(15,17)(16,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^3,3,2^3|b^2=g,c^2=g,e^2=g,f^2=g,d^a=d^2,e^a=f,f^a=e,c^b=cg,e^d=fg,f^d=ef,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #85466, #85467, #85469, … |
| maximal normal subgroup of | #85466, #85467, #85469, … |
| maximal quotient of | #85360, #85361, #85362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1483);