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) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C23 |
| Fitting | C3×#95 | C2 |
| hypercenter | #95 | D3 |
| hypocenter | C3 | #400 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #95 | |
134 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5)(6,7)(9,13)(12,15)(16,18,19,23)(17,21,20,22);(4,5,6,7)(8,10,11,14)(9,12,13,15)(17,20)(18,23);(2,3)(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,20)(18,22,23,21)Polycyclic presentation:
2^6,3|a^2=f,b^2=d,c^2=e,b^a=be,c^a=cde,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4658, #73351, #73352, … |
| maximal normal subgroup of | C2×#4658, #73351, #73352, … |
| maximal quotient of | #68810, #68811, #68812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 217);