#72236332

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

#45439 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#4625⋊C₂
commutator, cosocle	C₂×#120⋊C₃	C₂³
Frattini	C₂⁴	C₂²×S₄
Fitting	#45439	D₃
socle	C₂³	C₂×GL(2,ℤ/4)
hypercenter	C₂²	#4625⋊C₂
hypocenter	C₂²⋊A₄	#120
LCS	G ▷ C₂×#120⋊C₃ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂ ◁ C₂²

2138 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 2138 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 20
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 20 points:

(5,6,12,7)(8,11,18,17)(9,13,20,14)(10,19,15,16);(2,4)(6,14,12,10,7,19)(8,9,16)(11,17,20,13,15,18);(1,2)(3,4)

Polycyclic presentation:

2^5,3,2^4|b^a=be,c^b=cd,f^c=f^2,g^b=gi,g^c=h,g^d=gij,g^f=h,h^b=hj,h^c=g,h^d=hi,h^f=gh,i^c=ij,i^f=j,j^f=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72236671

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