Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator | C22×C3 | C22×C4 |
| Fitting | C2×C3×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C3 | C23 |
| hypercenter | C2×C4⋊C4 | D3 |
| hypocenter | C3 | #389 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
170 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 170 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) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(9,11)(12,13,14,15);(8,9,10,11);(2,3)(4,5,6,7)(13,15)Polycyclic presentation:
2^6,3|a^2=d,b^2=f,c^2=e,c^a=ce,g^a=g^2,c^b=cf48 conjugacy classes — view the full character table.
| maximal subgroup of | #68807, #68808, #68809, … |
| maximal normal subgroup of | #68807, #68808, #68809, … |
| maximal quotient of | #69407, #69412, #69419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 543);