#72243603

order	11760
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

derived, Hall	C₅×C₇² ⋊ C₃×C₄²
Hall	Dic₇×C₇⋊AGL(1,𝔽₅) ⋊ C₃
Hall	C₅×#3861 ⋊ C₄²
Hall	#3861 ⋊ C₄×AGL(1,𝔽₅)
Sylow	C₇² ⋊ C₃×C₄×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #72243526

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72219856
commutator, hypocenter	C₅×C₇²	C₃×C₄²
Fitting, socle	C₂×C₅×C₇²	C₂×C₃×C₄
prime radical	C₂×D₅×#3861	C₂²
cosocle	C₂×C₇²×D₅	C₂²×C₃
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ C₂

336 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11924174, #72243685, #72243694
maximal normal subgroup of	AGL(1,𝔽₅)×#11924174, #72243685, #72243694

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