#72228992

order	4000
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅³ ⋊ #95

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#14522937
commutator, Fitting, socle	C₂×C₅³	C₄²
prime radical, cosocle	C₂×D₅×C₅⋊D₅	C₂²
hypocenter	C₅³	#95
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂

508 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 19 points:

(2,4)(5,6,8,7)(11,12,13,14)(15,16,19,18);(1,2)(3,4)(6,7,8,9)(10,11,14,12,13)(16,17,19,18)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259230, #72259312
maximal normal subgroup of	#72259230, #72259312

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