Finite Group DatabasebetaFinite Group Databasebeta| order | 8096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×D4×C23 ⋊ C2×C11 |
| Hall | #10625155 ⋊ C11 |
| Hall | C23⋊C11 ⋊ H(5,𝔽2) |
| Sylow | C23 ⋊ C11×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×AGL(1,𝔽23) |
| commutator, socle, cosocle | C2×C23 | C24×C11 |
| Fitting | C2×D4×C23 | C2×C11 |
| hypercenter | C2×D4 | AGL(1,𝔽23) |
| hypocenter | C23 | C11×H(5,𝔽2) |
| LCS | G ▷ C2×C23 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
332 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 184 |
Polycyclic presentation:
2^4,11,2,23|d^2=f,b^a=bf,c^a=cf,d^a=df,d^b=df,g^a=g^22,g^e=g^18197 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×AGL(1,𝔽23), #72247935, #72247938, … |
| maximal normal subgroup of | H(5,𝔽2)×AGL(1,𝔽23), #72247935, #72247938, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 11, 2, 23]);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> SetConjugate(coll, 7, 5, F.7^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);