Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×C4⋊Q8 ⋊ C2 |
| Sylow | C7 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C4⋊Q8 | C2 |
| prime radical | C2×D28 | C22 |
| hypercenter | C4⋊Q8 | D7 |
| hypocenter | C7 | #563 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,4,6,7,5,3)(8,10,11,15)(9,13,12,14)(17,23)(21,29)(22,31)(28,30);(9,12)(13,14)(16,19,20,27)(17,22,23,31)(18,24,25,26)(21,28,29,30);(10,15)(13,14)(16,18,20,25)(17,21,23,29)(19,26,27,24)(22,30,31,28);(2,3)(4,5)(6,7)(8,9)(10,14)(11,12)(13,15)(16,17)(18,21)(19,22)(20,23)(24,28)(25,29)(26,30)(27,31)Polycyclic presentation:
2^6,7|b^2=e,c^2=e,d^2=f,c^a=cf,d^a=def,g^a=g^6,c^b=ce,d^b=df61 conjugacy classes — view the full character table.
| maximal subgroup of | #11726921, #11728747, C2×#90426, … |
| maximal normal subgroup of | #11726921, #11728747, C2×#90426, … |
| maximal quotient of | #11731792, #11731793, #11731794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1188);