Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#73026 (order 16128) · Inn(G) = C23×D7 · Out(G) = C2×C3×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C7×Q8 ⋊ C2 |
| Sylow | C7 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D7 |
| commutator, socle, cosocle | C2×C7 | C24 |
| Fitting | C2×C7×Q8 | C2 |
| hypercenter | C2×Q8 | D7 |
| hypocenter | C7 | 2−1+4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(9,15)(13,21)(14,23)(20,22);(8,11,12,19)(9,14,15,23)(10,16,17,18)(13,20,21,22);(8,10,12,17)(9,13,15,21)(11,18,19,16)(14,22,23,20);(2,3)(4,5)(6,7)(8,9)(10,13)(11,14)(12,15)(16,20)(17,21)(18,22)(19,23)Polycyclic presentation:
2^5,7|b^2=e,c^2=e,d^a=de,f^a=f^6,c^b=ce47 conjugacy classes — view the full character table.
| maximal subgroup of | #89524, #89525, #89834, … |
| maximal normal subgroup of | #89524, #89525, #89834, … |
| maximal quotient of | #90191, #90195, #90198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 183);