Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4656 |
| automorphism | Aut(G) = C29×D3 (order 3072) · Inn(G) = C22×D3 · Out(G) = C27 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #394 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator, socle, cosocle | C23×C3 | C23 |
| Fitting | C2×C3×C22⋊C4 | C2 |
| prime radical | C22×#67 | C2 |
| hypercenter | C2×C22⋊C4 | D3 |
| hypocenter | C3 | #394 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
238 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 61 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)(9,11)(12,14,15,13);(8,9)(10,11)(12,13)(14,15);(2,3)(4,5)(6,7)(13,14)Polycyclic presentation:
2^6,3|c^2=e,b^a=be,c^a=cd,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #73481, #73485, #73486, … |
| maximal normal subgroup of | #73481, #73485, #73486, … |
| maximal quotient of | #69409, #69414, #69428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 518);