Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C28×D3 (order 1536) · Inn(G) = C23×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C22⋊Q8 | C2 |
| prime radical | C2×#67 | C22 |
| hypercenter | C22⋊Q8 | D3 |
| hypocenter | C3 | #563 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,15,16,23)(13,19,18,26)(14,21,20,22)(17,24,25,27);(6,10)(8,11)(13,18)(14,20)(19,26)(21,22);(4,6)(5,8)(7,10)(9,11)(12,14,16,20)(13,17,18,25)(15,22,23,21)(19,27,26,24);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,17)(15,19)(16,18)(20,25)(21,24)(22,27)(23,26)Polycyclic presentation:
2^6,3|b^2=f,d^2=f,c^a=cf,d^a=def,g^a=g^2,c^b=ce,d^b=df33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5646, #85973, #85989, … |
| maximal normal subgroup of | C2×#5646, #85973, #85989, … |
| maximal quotient of | #73886, #73960, #73963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1205);