Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#6628842 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #499 |
86 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)(5,8,9,10)(13,18)(14,20)(15,22)(17,23)(21,27)(24,25);(12,14,15,20,16,21,22,27)(13,17,19,24,18,25,26,23);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,19)(16,18)(20,24)(21,25)(22,26)(23,27)Polycyclic presentation:
2^6,3|a^2=d,b^2=e,c^2=d,e^2=f,c^a=cdf,g^a=g^2,c^b=cef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69247, #69316, C2×#4858, … |
| maximal normal subgroup of | #69247, #69316, C2×#4858, … |
| maximal quotient of | #68804, #68828, #68836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 417);