Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×D4×AGL(1,ℤ/8) (order 1536) · Inn(G) = C2×D3×D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#135 ⋊ C2 |
| Sylow | C3 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×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 | #578 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 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) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,7,9,17)(5,12,10,24)(6,15,14,32)(8,21,22,19)(11,26,23,34)(13,31,25,29)(16,33,20,18)(27,35,30,28);(4,6,9,14)(5,11,10,23)(7,18,17,33)(8,20,22,16)(12,28,24,35)(13,30,25,27)(15,21,32,19)(26,31,34,29);(5,10)(6,14)(7,17)(13,25)(16,20)(19,21)(26,34)(28,35);(2,3)(4,5)(6,11)(7,12)(8,13)(9,10)(14,23)(15,26)(16,27)(17,24)(18,28)(19,29)(20,30)(21,31)(22,25)(32,34)(33,35)Polycyclic presentation:
2^6,3|c^2=f,d^2=f,e^2=f,b^a=bf,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 | #69243, #69245, #71518, … |
| maximal normal subgroup of | #69243, #69245, #71518, … |
| maximal quotient of | #78821, #78823, #78828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1330);