#72246748

order	7680
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×H(5,𝔽₂) ⋊ Dic₃
derived	C₅×C₂²⋊#997 ⋊ C₄
Sylow	C₅ ⋊ #72219111

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72246556
commutator, hypocenter	C₅×C₂²⋊#997	C₄
Fitting	C₂²×C₅×H(5,𝔽₂)	Dic₃
socle	C₂³×C₅	#5936
prime radical, cosocle	D₅×C₂²⋊#997	C₂
LCS	G ▷ C₅×C₂²⋊#997
UCS	C₁ ◁ C₂

14076 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 14076 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 17
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

2^2,3,5,2^7|a^2=b,c^a=c^2,d^a=d^3,d^b=d^4,e^a=ef,e^c=f,f^c=ef,g^a=gh,g^c=h,h^c=gh,i^a=ij,i^c=j,i^f=ik,j^c=ij,j^e=jk

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11715850
maximal normal subgroup of	AGL(1,𝔽₅)×#11715850
maximal quotient of	#72246787, #72258530

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);