Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72213256 (order 6144) · Inn(G) = C2×#67 · Out(G) = #1889 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #491 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,7,11)(5,9,8,10)(13,18)(14,21)(15,23)(17,24)(19,20)(25,26);(4,5,7,8)(6,10,11,9)(12,14)(13,17)(15,22)(16,21)(18,26)(19,24)(20,25)(23,27);(2,3)(5,8)(9,10)(12,13,16,20)(14,17,21,25)(15,18,23,19)(22,26,27,24)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=e,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,c^b=cef,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | #68374, #68377, #68379, … |
| maximal normal subgroup of | #68374, #68377, #68379, … |
| maximal quotient of | #69548, #69552, #69560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 628);