Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#117 | C2 |
| prime radical | C23×C3 | C23 |
| hypercenter | #117 | D3 |
| hypocenter | C3 | #541 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(4,6,7,10)(5,8,9,11)(15,23)(18,25)(21,27)(24,26);(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,16,19)(14,20,22,17)(15,18,23,25)(21,26,27,24)Polycyclic presentation:
2^6,3|a^2=f,b^2=f,c^2=e,d^2=f,b^a=bf,d^a=de,g^a=g^2,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #78567, #78570, #78577, … |
| maximal normal subgroup of | #78567, #78570, #78577, … |
| maximal quotient of | #73386, #73388, #73390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1077);