Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C7 ⋊ #408 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting, prime radical | C2×C7×#41 | C2 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | #408 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6,7,5,3)(9,11)(14,20)(17,23)(21,26)(24,27);(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^6,7|a^2=d,d^2=f,c^a=ce,g^a=g^6,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #11725502, #11725503, #11725506, … |
| maximal normal subgroup of | #11725502, #11725503, #11725506, … |
| maximal quotient of | #11729333, #11729336, #11729337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 768);