Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of #11924306 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle | C23×C7 ⋊ ΓL(1,𝔽8) |
| derived, Hall | C7×AGL(1,𝔽8) ⋊ C3 |
| Sylow | C23 ⋊ #3861 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×AGL(1,𝔽8) | C3 |
| Fitting, socle | C23×C7 | ΓL(1,𝔽8) |
| LCS | G ▷ C7×AGL(1,𝔽8) | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
3,7^2,2^3|b^a=b^4,c^a=c^4,d^a=ef,e^a=de,d^b=df,e^b=f,f^b=def24 conjugacy classes — view the full character table.
| maximal subgroup of | #72214179, ΓL(1,𝔽8)×AΓL(1,𝔽8), #72214688, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×AΓL(1,𝔽8), #72214688, C2×#11924306, … |
| maximal quotient of | #72214179, C2×#11924306, C3×#11924306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 221);