#72259458

order	16224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₁₃² ⋊ C₂×C₃×C₄
Hall	#72259238 ⋊ C₃
Hall	#92788 ⋊ C₂×C₄⋊C₄
Sylow	C₁₃² ⋊ C₂×C₃×C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#72210302
commutator, socle	C₂×C₁₃²	C₂²×C₃×C₄
Fitting	C₄×C₁₃²	C₂×C₃×C₄
prime radical	C₂×#11759815⋊C₂	C₂²
cosocle	C₂×C₁₃⋊D₁₃	C₂³×C₃
hypercenter	C₄	#72210302
hypocenter	C₁₃²	C₂×C₃×C₄⋊C₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

524 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^3,3,2^2,13^2|a^2=e,b^2=e,c^2=e,b^a=bf,c^b=cf,g^a=g^5,g^b=g^8,g^c=g^8,g^d=g^9,g^e=g^12,h^a=g^6h^8,h^b=h^8,h^c=h^8,h^d=h^9,h^e=h^12

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#72210302
maximal normal subgroup of	D₄×#72210302

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^5);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^8);;
gap> SetConjugate(coll, 7, 4, F.7^9);;
gap> SetConjugate(coll, 7, 5, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8^8);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 3, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^9);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);