Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1141 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C22⋊D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#41 ⋊ C2 |
| Sylow | C7 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊D7 |
| commutator, cosocle | C4×C7 | C23 |
| Frattini | C4 | C22×D7 |
| Fitting | C7×#41 | C2 |
| socle | C2×C7 | C2×D4 |
| hypercenter | #41 | D7 |
| hypocenter | C7 | #135 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
62 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)(9,14)(11,19)(13,20)(15,16)(17,23)(21,22);(8,10,12,18)(9,13,16,22)(11,17,19,23)(14,20,15,21);(2,3)(4,5)(6,7)(8,9)(10,13)(11,14)(12,16)(15,19)(17,20)(18,22)(21,23)Polycyclic presentation:
2^5,7|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^6,d^c=de44 conjugacy classes — view the full character table.
| maximal subgroup of | #89595, #89596, #89916, … |
| maximal normal subgroup of | #89595, #89596, #89916, … |
| maximal quotient of | #89768, #89770, #89774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 145);