Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of #11716142 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Fitting, socle, Sylow | C72 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | #5997 | C22 |
| Fitting, socle, hypocenter | C72 | Q16 |
| prime radical | C7⋊D7 | D4 |
| LCS | G ▷ #5997 ▷ C7⋊D7 ▷ C72 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 49 |
Polycyclic presentation:
2^4,7^2|a^2=cd,b^2=d,c^2=d,b^a=bcd,e^a=e^4f^5,f^a=e^6f^6,c^b=cd,e^b=e^4f^4,f^b=ef^3,e^c=e^3f^6,f^c=e^3f^4,e^d=e^6,f^d=f^6| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 16 | 16 | 16 | 49 | 98 | 98 | 98 | 196 | 196 |
| order | 1 | 7 | 7 | 7 | 2 | 4 | 8 | 8 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 |
| χ5 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 |
| χ6 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | ||
| χ7 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | ||
| χ8 | 16 | −2ζ12+ζ20+ζ36−2ζ44 | ζ4−2ζ20−2ζ36+ζ52 | −2ζ4+ζ12+ζ44−2ζ52 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ9 | 16 | ζ4−2ζ20−2ζ36+ζ52 | −2ζ4+ζ12+ζ44−2ζ52 | −2ζ12+ζ20+ζ36−2ζ44 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 16 | −2ζ4+ζ12+ζ44−2ζ52 | −2ζ12+ζ20+ζ36−2ζ44 | ζ4−2ζ20−2ζ36+ζ52 | 0 | 0 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 56 (ζ56 = 1).
| maximal subgroup of | #11716140⋊C2, C2×#11716142, #72209731, … |
| maximal normal subgroup of | #11716140⋊C2, C2×#11716142, #72224473 |
| maximal quotient of | #87658⋊C4, #13096625, C2×#11716142 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 162);