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 | C7×#95 | C2 |
| hypercenter | #95 | D7 |
| hypocenter | C7 | #400 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #95 | |
134 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(8,9,10,11)(13,17)(16,19)(22,27)(24,26);(9,11)(12,14,15,18)(13,16,17,19)(20,22)(21,24)(23,27)(25,26);(2,3)(4,5)(6,7)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,26,27,24)Polycyclic presentation:
2^6,7|a^2=e,b^2=d,c^2=ef,b^a=be,c^a=cd,g^a=g^6,c^b=cf82 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89432, #11730795, #11730797, … |
| maximal normal subgroup of | C2×#89432, #11730795, #11730797, … |
| maximal quotient of | #11726505, #11726596, #11726597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 194);