Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63419 |
| automorphism | Aut(G) = (#63419⋊C2)⋊C3 (order 1728) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C42⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42⋊C9 | C2 |
| Frattini, socle | C22×C3 | S4 |
| Fitting | C3×C42 | D3 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,3,4,8)(2,12,11,10,6,9,7,5)(14,16)(15,19)(17,18)(20,21);(1,2,5)(3,7,9)(4,6,10)(8,11,12)(13,21,16,19,18,17,15,14,20)Polycyclic presentation:
2,3^2,2^4|b^3=c^2,d^2=fg,e^2=f,b^a=b^2c,c^a=c^2,d^a=e,e^a=d,f^a=fg,d^b=e,e^b=deg,f^b=g,g^b=fg18 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63419, #63419⋊C2, #63419⋊C3, … |
| maximal normal subgroup of | C2×#63419, #63419⋊C2, #63419⋊C3, … |
| maximal quotient of | #10588301, #10588302, #10588303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 67);