#72276963

order	3248
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Hall	#90777 ⋊ C₇
Hall	C₂₉⋊C₇ ⋊ M₁₆
Sylow	C₂₉ ⋊ C₇×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂×D₂₉⋊C₇
commutator, socle	C₂×C₂₉	C₂×C₄×C₇
Fitting	C₂×C₄×C₂₉	C₂×C₇
prime radical	C₂²×C₂₉⋊C₇	C₄
cosocle	C₄×C₂₉	C₂²×C₇
hypercenter	C₂×C₄	D₂₉⋊C₇
hypocenter	C₂₉	C₇×M₁₆
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 232

Polycyclic presentation:

2^2,7,2^2,29|a^2=d,d^2=e,b^a=be,f^a=f^28,f^c=f^16

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	M₁₆×D₂₉⋊C₇
maximal normal subgroup of	M₁₆×D₂₉⋊C₇

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, 2, 29]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 6, 3, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);