Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4656 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator, socle, cosocle | C23×C3 | C23 |
| Fitting | C22×C3×D4 | C2 |
| hypercenter | C22×D4 | D3 |
| hypocenter | C3 | #392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
322 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(13,15);(9,11)(12,13)(14,15);(2,3)(4,5)(6,7)(8,9)(10,11)Polycyclic presentation:
2^6,3|b^a=be,c^a=cd,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #70968, #70969, #70976, … |
| maximal normal subgroup of | #70968, #70969, #70976, … |
| maximal quotient of | #71547, #71550, #71568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 782);