#72244990

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived, Sylow	C₇² ⋊ C₂×C₃×C₈
Hall	#11716029 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₇×AGL(1,𝔽₇)
commutator, hypocenter	C₇²	C₂×C₃×C₈
Fitting, cosocle	C₄×C₇²	C₂²×C₃
socle	C₂×C₇²	C₂×C₃×C₄
prime radical	C₂×ΓL(1,𝔽₈)⋊D₇	C₄
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^2,3,2^2,7^2|a^2=d,d^2=e,f^a=f^6,f^b=f^6,f^c=f^4,g^a=f^6g,g^b=g^6,g^c=f^5g

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244996, #72245024, #72245032, …
maximal normal subgroup of	#72244996, #72245024, #72245032, …

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.6^5*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);