#72254144

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ M₁₆
Sylow	C₇ ⋊ C₃×#3712

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×C₃⋊AGL(1,𝔽₇)
commutator, socle	C₂×C₃²×C₇	C₂×C₃×C₄
Fitting, cosocle	C₃²×C₄×C₇	C₂²×C₃
prime radical	C₂×C₃⋊(C₃⋊AGL(1,𝔽₇))	C₄
hypocenter	C₃²×C₇	C₃×M₁₆
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₄

236 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 54 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:

2^2,3,2^2,3^2,7|a^2=d,d^2=e,b^a=be,f^a=f^2,f^b=f^2,g^a=f^2g,g^b=g^2,h^b=h^6,h^c=h^4

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254145, #72254160, AGL(1,𝔽₇)×#3712, …
maximal normal subgroup of	#72254145, #72254160, AGL(1,𝔽₇)×#3712, …

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