#72244492

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅×H(5,𝔽₂) ⋊ C₄×D₃
derived	C₅×#997 ⋊ C₂×C₄
Sylow	C₅ ⋊ #11711591

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#1020⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×#997	C₂×C₄
Frattini	C₂³	AGL(1,𝔽₅)×S₄
Fitting	C₅×H(5,𝔽₂)	C₄×D₃
socle	C₂×C₅	#1020⋊C₄
prime radical	D₅×#5935	C₂
cosocle	D₅×#997	C₂²
LCS	G ▷ C₅×#997
UCS	C₁ ◁ C₂

832 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

(1,6,3,2)(4,8,7,5)(9,10,12,11);(2,7,6,5,3,8)(10,11,12,13)

Polycyclic presentation:

2^3,3,5,2^5|a^2=c,b^2=c,d^b=d^2,e^a=e^3,e^b=e^3,e^c=e^4,f^a=fij,f^b=fg,f^d=g,g^a=ghj,g^d=fg,h^d=hi,h^f=hj,i^b=hi,i^d=h,i^g=ij

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#73026
maximal normal subgroup of	AGL(1,𝔽₅)×#73026

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