#72341284

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#379 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#1020⋊C₂
commutator, cosocle	C₂²×C₂²⋊A₄	C₂³
Frattini, socle	C₂⁴	C₂²×S₄
Fitting	C₂²×#379	D₃
hypercenter	C₂³	#1020⋊C₂
hypocenter	C₂²⋊A₄	#121
LCS	G ▷ C₂²×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂² ◁ C₂³

5319 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 5319 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341296
maximal normal subgroup of	#72341296
maximal quotient of	#72341378

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, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.7);;
gap> SetConjugate(coll, 9, 1, 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);