#72258493

order	9600
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅² ⋊ C₄×D₃
derived	C₅²×C₂²⋊A₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ #1020⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₂²⋊A₄	C₂×C₄
Frattini	C₂²	AGL(1,𝔽₅)×C₅⋊S₄
Fitting	C₂⁴×C₅²	C₄×D₃
socle	C₂²×C₅²	C₄×S₄
prime radical	D₅×C₅⋊#1020	C₂
cosocle	C₅×D₅×C₂²⋊A₄	C₂²
LCS	G ▷ C₅²×C₂²⋊A₄
UCS	C₁

1020 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 18
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2^3,3,5^2,2^4|a^2=c,d^b=d^2,e^a=e^2,e^b=e^4,e^c=e^4,f^a=ef,f^b=f^4,f^c=e^3f,g^a=gi,g^b=gh,g^d=h,h^a=hij,h^d=gh,i^b=j,i^d=ij,j^b=i,j^d=i

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#1020⋊AGL(1,𝔽₅)
maximal normal subgroup of	D₅×#1020⋊AGL(1,𝔽₅)

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 5, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 5, 3, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^4);;
gap> SetConjugate(coll, 6, 3, F.5^3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);