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 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,13,18,11,4,29,6,25,16,3,17,30)(2,19,24,12,8,31,10,27,22,7,23,32)(5,14,15,26)(9,20,21,28);(1,2,6,10)(3,12,11,7)(4,8,17,23)(5,9,15,21)(13,27,25,19)(14,28,26,20)(16,22,18,24)(29,32,30,31);(3,11)(4,5)(7,12)(8,9)(13,26)(14,25)(15,17)(16,18)(19,28)(20,27)(21,23)(22,24)Polycyclic presentation:
2^3,3,2^3|b^2=g,c^2=g,e^2=g,f^2=g,c^a=cg,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 | #85467, #85468, #85470, … |
| maximal normal subgroup of | #85467, #85468, #85470, … |
| maximal quotient of | #85363, #85365, #85366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1484);