#72243093

order	2800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₅²×C₇ ⋊ C₂²⋊C₄
Sylow	C₅² ⋊ C₂²⋊Dic₇
Sylow	C₇ ⋊ #87831

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅⋊(C₇⋊AGL(1,𝔽₅))
commutator, Fitting, socle	C₂×C₅²×C₇	C₂×C₄
prime radical	C₂×C₇×D₅²	C₂
cosocle	C₂×C₇×C₅⋊D₅	C₂²
hypocenter	C₅²×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₅²×C₇ ▷ C₅²×C₇
UCS	C₁ ◁ C₂

168 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 35 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:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243094, #72243105, D₇×#87831, …
maximal normal subgroup of	#72243094, #72243105, D₇×#87831, …

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