Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×#67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #459 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#67 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×C4×D4 | C2 |
| socle | C22×C3 | C2×D4 |
| hypercenter | C4×D4 | D3 |
| hypocenter | C3 | #459 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D4 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(9,12)(10,15)(13,14);(4,5,6,7);(2,3)(5,7)(8,9)(10,12)(11,14)(13,15)Polycyclic presentation:
2^6,3|b^2=e,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #68180, #68186, #68187, … |
| maximal normal subgroup of | #68180, #68186, #68187, … |
| maximal quotient of | #69462, #69466, #69487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 574);