Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#3003 (order 768) · Inn(G) = D3×D4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#135 ⋊ C2 |
| Sylow | C3 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D3×D4 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×#135 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | #135 | D3 |
| hypocenter | C3 | #575 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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)(4,6)(5,10)(7,16)(8,12)(9,13)(11,18)(14,15)(17,19);(5,9)(6,14)(7,15)(10,18)(12,16)(13,19);(4,5)(6,13)(7,11)(8,9)(10,12)(14,18)(15,17)(16,19);(2,3)(6,12)(10,13)(14,16)(18,19)Polycyclic presentation:
2^6,3|e^2=f,d^a=df,g^a=g^2,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69252, #69253, #71516, … |
| maximal normal subgroup of | #69252, #69253, #71516, … |
| maximal quotient of | #78819, #78822, #78826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1327);