Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = GL(2,ℤ/4) · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | H(5,𝔽2) | D3 |
| hypercenter | Q8 | S4 |
| hypocenter | SL(2,𝔽3) | D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,15,24,6,11,21,7,13,17,2,19,20)(3,16,18,8,14,12,10,9,23,5,4,22);(2,8)(3,10)(4,12)(5,6)(9,17)(11,20)(13,18)(14,22)(15,23)(16,24)(19,21)Polycyclic presentation:
2^3,3,2^3|b^2=g,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,c^b=cg,e^d=fg,f^d=ef,f^e=fg17 conjugacy classes — view the full character table.
| maximal subgroup of | #85467, #85468, #85469, … |
| maximal normal subgroup of | #85467, #85468, #85469, … |
| maximal quotient of | #73076, #73078, #73090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 988);