#72245808

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₇ ⋊ C₂×SD₁₆
Sylow	C₃² ⋊ D₇×SD₁₆
Sylow	C₇ ⋊ #63930

Subgroups

	subgroup	G/H
center	C₂	#11759626
commutator, cosocle	C₃²×C₄×C₇	C₂³
Frattini	C₄	C₂×D₃×D₂₁
Fitting	C₃²×C₇×D₄	C₂²
socle	C₂×C₃²×C₇	C₂×D₄
prime radical	D₄×C₃⋊D₂₁	C₂
hypercenter	D₄	D₃×D₂₁
hypocenter	C₃²×C₇	C₂×SD₁₆
LCS	G ▷ C₃²×C₄×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

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

normal subgroups only

This group has 326 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

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^5,3^2,7|a^2=d,d^2=e,b^a=bde,d^b=de,f^a=f^2,f^c=f^2,g^a=f^2g,g^c=g^2,h^c=h^6

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#63930, #72245926, #72245935, …
maximal normal subgroup of	D₇×#63930, #72245926, #72245935, …

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