#72243181

order	4200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	#13063789 ⋊ C₃
Hall	C₅²×ΓL(1,𝔽₈) ⋊ D₄
Hall	C₅²×C₇ ⋊ C₃×D₄
Sylow	C₅² ⋊ C₄⋊AGL(1,𝔽₇)
Hall	ΓL(1,𝔽₈) ⋊ #6038
Sylow	C₇ ⋊ C₃×#6038

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×C₅⋊AGL(1,𝔽₇)
commutator, Fitting, socle, cosocle	C₂×C₅²×C₇	C₂²×C₃
hypocenter	C₅²×C₇	C₃×D₄
LCS	G ▷ C₂×C₅²×C₇ ▷ C₅²×C₇
UCS	C₁ ◁ C₂

168 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 140

Polycyclic presentation:

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

Conjugacy classes & character table

127 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243182, #72243211, AGL(1,𝔽₇)×#6038
maximal normal subgroup of	#72243182, #72243211, AGL(1,𝔽₇)×#6038

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