Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×S4 (order 2304) · Inn(G) = C22×#67 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#67 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×H(5,𝔽2) | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | H(5,𝔽2) | D3 |
| hypocenter | C3 | #576 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,10)(7,15)(9,17)(11,12)(14,19)(16,18);(6,13)(9,16)(14,19)(17,18);(4,6)(5,9)(7,14)(8,13)(10,17)(11,18)(12,16)(15,19);(2,3)(4,5)(6,9)(7,10)(8,12)(11,15)(13,16)(14,17)(18,19)Polycyclic presentation:
2^6,3|e^2=f,d^a=de,e^a=ef,g^a=g^2,c^b=cf,e^d=ef39 conjugacy classes — view the full character table.
| maximal subgroup of | #80280, #80282, #80294, … |
| maximal normal subgroup of | #80280, #80282, #80294, … |
| maximal quotient of | #82537, #82538, #82539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1394);