#72257984

order	47096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow	C₂₉² ⋊ C₂×C₄×C₇
Hall	#72257983 ⋊ C₇
Hall	#19154634 ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₂₉²	C₂×C₄×C₇
prime radical	#26044455⋊C₂	C₂
cosocle	C₂₉⋊D₂₉	C₂²×C₇
LCS	G ▷ C₂₉²
UCS	C₁

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 58

Polycyclic presentation:

2^2,7,2,29^2|b^2=d,e^a=e^28,e^b=e^17,e^c=e^24,e^d=e^28,f^a=e^12f,f^b=f^17,f^c=f^24,f^d=f^28

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

holomorph of

D₂₉

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 7, 2, 29, 29]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> SetConjugate(coll, 5, 2, F.5^17);;
gap> SetConjugate(coll, 5, 3, F.5^24);;
gap> SetConjugate(coll, 5, 4, F.5^28);;
gap> SetConjugate(coll, 6, 1, F.5^12*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^17);;
gap> SetConjugate(coll, 6, 3, F.6^24);;
gap> SetConjugate(coll, 6, 4, F.6^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);