Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#103 ⋊ C2 |
| Sylow | C3 ⋊ #416 |
158 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 158 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) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7)(8,11,12,10,13,19,21,20)(9,15,16,18,17,22,23,14);(8,10,13,20)(9,14,17,18)(11,21,19,12)(15,16,22,23);(2,3)(8,9)(10,18)(11,15)(12,16)(13,17)(14,20)(19,22)(21,23)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,d^2=f,e^2=f,b^a=bf,g^a=g^2,c^b=ce,d^b=df,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4804, #78790, #78807, … |
| maximal normal subgroup of | C2×#4804, #78790, #78807, … |
| maximal quotient of | #68229, #68233, #68237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 363);