Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×2−1+4 ⋊ C2 |
| Sylow | C7 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6516 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×2−1+4 | C2 |
| socle | C2×C7 | #120 |
| hypercenter | 2−1+4 | D7 |
| hypocenter | C7 | #454 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ 2−1+4 | |
146 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)(8,10,12,17)(9,14,15,23)(11,20,18,16)(13,21,19,22);(8,9)(10,14)(11,19)(12,15)(13,18)(16,21)(17,23)(20,22);(2,3)(4,5)(6,7)(9,13)(11,18)(14,22)(15,19)(16,20)(21,23)Polycyclic presentation:
2^6,7|c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^6,d^b=df,e^b=ef,e^c=ef67 conjugacy classes — view the full character table.
| maximal subgroup of | #11728741, #11728742, #11728747, … |
| maximal normal subgroup of | #11728741, #11728742, #11728747, … |
| maximal quotient of | #11729412, #11729415, #11729419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 779);