Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×#3003 (order 3072) · Inn(G) = D3×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C4×Q16 |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,6,7,5)(10,16)(11,19)(13,20)(14,23)(17,18)(21,22);(4,5,7,6)(8,10)(9,13)(11,16)(12,18)(14,20)(15,22)(17,19)(21,23);(2,3)(4,5,7,6)(8,9,12,15)(10,13,18,22)(11,14,19,23)(16,20,17,21)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=ef,d^2=f,g^a=g^2,c^b=cd,d^b=df,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #69175, #69176, #69204, … |
| maximal normal subgroup of | #69175, #69176, #69204, … |
| maximal quotient of | #68557, #68560, #68564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 432);