#72259348

order	5000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₅

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₅⁴ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅⁴	C₂×C₄
prime radical	C₅⋊D₅²	C₂
cosocle	C₅⋊(C₅⋊(C₅⋊D₅))	C₂²
LCS	G ▷ C₅⁴
UCS	C₁

1056 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 100

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259565, #1043×#1044, #72259604
maximal normal subgroup of	#72259565, #1043×#1044, #72259604

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