Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #6400 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#442 (order 2688) · Inn(G) = #1141 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #1141 |
| commutator, cosocle | C7×C8 | C22 |
| Frattini | C8 | C2×D7 |
| Fitting | C7×Q16 | C2 |
| socle, prime radical | C2×C7 | D8 |
| hypercenter | Q16 | D7 |
| hypocenter | C7 | Q32 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,4,6,7,5,3)(8,10,13,21)(9,14,17,31)(11,23,26,39)(12,28,24,20)(15,33,34,38)(16,35,27,30)(18,36,37,22)(19,32,29,25);(2,3)(4,5)(6,7)(8,9,13,17)(10,18,21,37)(11,22,26,36)(12,27,24,16)(14,29,31,19)(15,32,34,25)(20,38,28,33)(23,35,39,30)Polycyclic presentation:
2^5,7|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,c^b=cd,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #89687, #89688, D7×Q32, … |
| maximal normal subgroup of | #89687, #89688, D7×Q32, … |
| maximal quotient of | #89284, #89287, #89359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 35);