Finite Group DatabasebetaFinite Group Databasebeta| order | 147 |
| isoclinism | isoclinic class of #3862 |
| automorphism | Aut(G) = #72241578 (order 3528) · Inn(G) = G · Out(G) = C3×D4 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C72 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72 | C3 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3,4)(5,7,6)(9,10,11)(12,14,13);(1,2,3,5,4,6,7)(8,9,11,13,10,12,14)Polycyclic presentation:
3,7^2|b^a=b^2,c^a=c^419 conjugacy classes — view the full character table.
| derived subgroup of | #64418, #10596868, #10596885, … |
| maximal subgroup of | #64418, #64422, #64425, … |
| maximal normal subgroup of | #64418, #64422, #64425, … |
| maximal quotient of | C2×#3862, #89205, C3×#3862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(147, 5);