Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×H(5,𝔽2) ⋊ C2 |
| Sylow | C7 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6516 |
| commutator, cosocle | C23×C7 | C23 |
| Frattini | C23 | C22×D7 |
| Fitting | C7×H(5,𝔽2) | C2 |
| socle | C2×C7 | #120 |
| hypercenter | H(5,𝔽2) | D7 |
| hypocenter | C7 | B(4,𝔽2) |
| LCS | G ▷ C23×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ H(5,𝔽2) | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,6,7,5,3)(8,10)(9,12)(11,14)(13,15);(8,9)(10,12)(11,13)(14,15);(2,3)(4,5)(6,7)(12,15)Polycyclic presentation:
2^6,7|b^a=be,c^a=cd,g^a=g^6,d^b=df,e^c=ef67 conjugacy classes — view the full character table.
| maximal subgroup of | #11728738, #11728739, #11728744, … |
| maximal normal subgroup of | #11728738, #11728739, #11728744, … |
| maximal quotient of | #11729408, #11729411, #11729599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 778);