Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = #953 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #953 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×H(5,𝔽2) | C2 |
| socle | C2×C3 | #120 |
| hypercenter | H(5,𝔽2) | D3 |
| hypocenter | C3 | #453 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2) | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2,3)(6,10)(8,11);(4,5)(6,11)(7,9)(8,10);(2,3)(5,8)(6,10)(9,11)Polycyclic presentation:
2^6,3|e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^2,d^b=df,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #70996, #70999, #71001, … |
| maximal normal subgroup of | #70996, #70999, #71001, … |
| maximal quotient of | #71647, #71650, #71664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 800);