Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = D4×D7 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×Q16 ⋊ C2 |
| Sylow | C7 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D7 |
| commutator, cosocle | C4×C7 | C23 |
| Frattini | C4 | C22×D7 |
| Fitting | C7×Q16 | C2 |
| socle | C2×C7 | C2×D4 |
| prime radical | #1161 | C2 |
| hypercenter | Q16 | D7 |
| hypocenter | C7 | #137 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,12,16)(9,13,15,22)(11,20,21,18)(14,23,19,17);(8,9,12,15)(10,17,16,23)(11,19,21,14)(13,20,22,18);(2,3)(4,5)(6,7)(10,16)(13,22)(17,23)(18,20)Polycyclic presentation:
2^5,7|b^2=e,c^2=e,d^2=e,c^a=ce,f^a=f^6,c^b=cd,d^b=de,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6481, #90457, #90459, … |
| maximal normal subgroup of | C2×#6481, #90457, #90459, … |
| maximal quotient of | #89561, #89563, #89567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 113);