#72341537

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

#14680 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#1020⋊C₂
commutator	C₂×C₂²⋊A₄	C₂²×C₄
Frattini, socle	C₂⁴	C₂²×S₄
Fitting	#14680	D₃
prime radical	C₂×C₄⋊#1020	C₂
cosocle	C₂²×C₂²⋊A₄	C₂³
hypercenter	C₂×C₄	#1020⋊C₂
hypocenter	C₂²⋊A₄	C₄×D₄
LCS	G ▷ C₂×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂² ◁ C₂×C₄

2587 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 2587 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 16
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#11713271, #72341613, D₄×#1020⋊C₄
maximal normal subgroup of	C₄×#11713271, #72341613, D₄×#1020⋊C₄
maximal quotient of	#72341764, #72341861, #72342132

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