#72231215

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₅ ⋊ #624
derived	C₅×A₄² ⋊ C₂×C₄
Sylow	C₅ ⋊ #11923458

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅×A₄²	C₂×C₄
Fitting, socle	C₂⁴×C₅	#624
prime radical	D₅×A₄⋊S₄	C₂
cosocle	D₅×A₄²	C₂²
LCS	G ▷ C₅×A₄²
UCS	C₁

1030 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 80

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×S₄², #72260445
maximal normal subgroup of	AGL(1,𝔽₅)×S₄²
maximal quotient of	#72260445

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