#72243131

order	8400
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

derived, Hall	C₅²×C₇ ⋊ C₃×C₄²
Hall	#72243056 ⋊ C₃
Hall	C₅²×ΓL(1,𝔽₈) ⋊ C₄²
Sylow	C₅² ⋊ C₄×ΓL(1,𝔽₈)⋊C₄
Hall	ΓL(1,𝔽₈) ⋊ #87830
Sylow	C₇ ⋊ C₃×#87830

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72218913
commutator, hypocenter	C₅²×C₇	C₃×C₄²
Fitting, socle	C₂×C₅²×C₇	C₂×C₃×C₄
prime radical	C₂×ΓL(1,𝔽₈)×C₅⋊D₅	C₂²
cosocle	C₂×C₇×C₅⋊D₅	C₂²×C₃
LCS	G ▷ C₅²×C₇
UCS	C₁ ◁ C₂

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#87830, #72243219, #72243223
maximal normal subgroup of	AGL(1,𝔽₇)×#87830, #72243219, #72243223

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 5, 5, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);