Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #857 |
| automorphism | Aut(G) = C2×#857⋊C2 (order 384) · Inn(G) = #857 · Out(G) = C22 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C42⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #857 |
| commutator, hypocenter | C42⋊C3 | C4 |
| Frattini, socle | C23 | S4 |
| Fitting | C2×C42 | D3 |
| prime radical, cosocle | C2×C42⋊C3 | C2 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(4,11,10,8,7,9,6,12);(1,4,9)(2,6,8)(3,7,11)(5,10,12)Polycyclic presentation:
2^2,3,2^4|a^2=b,d^2=fg,e^2=f,c^a=c^2,d^a=e,e^a=d,f^a=fg,d^c=e,e^c=deg,f^c=g,g^c=fg20 conjugacy classes — view the full character table.
| maximal subgroup of | #72980, C4×#857, #72992, … |
| maximal normal subgroup of | #72980, C4×#857, #72992, … |
| maximal quotient of | #67995, SL(2,ℤ/8), #67997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 182);