Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C22×C8×AGL(1,𝔽7) (order 1344) · Inn(G) = C2×D7 · Out(G) = C2×C3×C8 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C32 ⋊ C2 |
| Sylow | C7 ⋊ M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×D7 |
| commutator, socle | C2×C7 | C2×C16 |
| Fitting | C7×C32 | C2 |
| prime radical | C2×D7 | C16 |
| cosocle | C7×C16 | C22 |
| hypercenter | C32 | D7 |
| hypocenter | C7 | M64 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C16 ◁ C32 | |
32 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)(8,9,10,15,11,16,18,27,12,17,19,28,21,29,30,37,13,14,20,24,22,25,31,34,23,26,32,35,33,36,38,39);(2,3)(4,5)(6,7)(9,14)(15,24)(16,25)(17,26)(27,34)(28,35)(29,36)(37,39)Polycyclic presentation:
2^6,7|b^2=c,c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^6136 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89242, #11727120, D7×M64, … |
| maximal normal subgroup of | C2×#89242, #11727120, D7×M64, … |
| maximal quotient of | #11725395, Dic7⋊C32, (C2×D7)⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 4);