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×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C4×D4 | C2 |
| hypercenter | C4×D4 | D3 |
| hypocenter | C3 | #553 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(12,15,16,14)(13,17,18,19);(5,8)(9,11)(12,14,16,15)(13,17,18,19);(6,10)(9,11);(2,3)(4,5,7,8)(6,9,10,11)(12,13,16,18)(14,19,15,17)Polycyclic presentation:
2^6,3|a^2=f,c^2=e,d^2=f,c^a=cf,g^a=g^2,d^b=def45 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5540, #85785, #85787, … |
| maximal normal subgroup of | C2×#5540, #85785, #85787, … |
| maximal quotient of | #73458, #73461, #73462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1099);