Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #89428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator, socle, prime radical, cosocle | C23×C7 | C23 |
| Fitting | C2×C7×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D7 |
| hypocenter | C7 | #400 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6,7,5,3)(9,11)(20,22,23,26)(21,24,25,27);(12,14,15,19)(13,16,17,18)(22,26)(24,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,17)(14,18,19,16)(20,21,23,25)(22,24,26,27)Polycyclic presentation:
2^6,7|a^2=e,b^2=ef,c^2=f,b^a=bef,c^a=cd,g^a=g^6,c^b=cf82 conjugacy classes — view the full character table.
| maximal subgroup of | #11725265, #11725302, #11725890, … |
| maximal normal subgroup of | #11725265, #11725302, #11725890, … |
| maximal quotient of | #11727169, #11727171, #11727177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 520);