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) = C23 |
No aliases for this group.
| 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 | #41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
173 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4)(6,10)(7,11);(5,7,8,11)(6,9,10,12);(2,3)(5,6)(7,9)(8,10)(11,12)Polycyclic presentation:
2^4,3,2^2|b^2=d,c^a=cd,e^a=e^2,f^a=g,g^a=f,c^b=cd,f^e=g,g^e=fg25 conjugacy classes — view the full character table.
| maximal subgroup of | #85457, #85458, #85461, … |
| maximal normal subgroup of | #85457, #85458, #85461, … |
| maximal quotient of | #85292, #85293, #85294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1473);