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.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | H(5,𝔽2) | D3 |
| prime radical | #994 | C2 |
| hypercenter | Q8 | S4 |
| hypocenter | SL(2,𝔽3) | D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,19,46,6,22,29,8,25,36,3,37,21)(2,11,41,27,30,35,13,32,38,9,26,45)(4,5,43,15,24,39,7,23,44,18,16,47)(10,31,42,14,33,40,17,28,20,12,34,48);(1,2,8,13)(3,14,6,12)(4,17,7,10)(5,20,23,42)(9,18,27,15)(11,29,32,21)(16,35,24,45)(19,38,25,41)(22,40,37,48)(26,43,30,44)(28,39,31,47)(33,36,34,46)Polycyclic presentation:
2^3,3,2^3|a^2=g,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 | #85466, #85467, #85471, … |
| maximal normal subgroup of | #85466, #85467, #85471, … |
| maximal quotient of | #73075, #73077, #73089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 987);