Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C22×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D4×S4 |
| Fitting | C22×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×S4 |
| commutator, cosocle | C2×A4 | C23 |
| Fitting | C22×D4 | D3 |
| socle | C23 | C22×D3 |
| hypercenter | D4 | S4 |
| hypocenter | A4 | C2×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
234 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 8 points:
(5,6,8,7);(2,4)(6,7,8);(1,2)(3,4)25 conjugacy classes — view the full character table.
| maximal subgroup of | D8×S4, #85457, SD16×S4, … |
| maximal normal subgroup of | D8×S4, #85457, SD16×S4, … |
| maximal quotient of | #85293, C22⋊C4×S4, GL(2,ℤ/4)⋊C4, … |
| automorphism group of | A4⋊Q8, C4⋊S4, C2×#226, … |
| outer automorphism group of | C2×#41, #520, D4×Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1472);