Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C7 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator | C22×C7 | C22×C4 |
| Fitting | C2×C7×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C7 | C23 |
| hypercenter | C2×C4⋊C4 | D7 |
| hypocenter | C7 | #387 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
154 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,9,10,11)(18,22)(20,23);(13,15)(16,18,19,22)(17,20,21,23);(2,3)(4,5)(6,7)(12,13)(14,15)(16,17,19,21)(18,20,22,23)Polycyclic presentation:
2^6,7|a^2=f,b^2=f,c^2=d,b^a=be,g^a=g^6,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #11731064, #11731065, C4×#6461, … |
| maximal normal subgroup of | #11731064, #11731065, C4×#6461, … |
| maximal quotient of | #11727156, #11727160, #11727172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 512);