#72224682

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₃²×C₇ ⋊ C₃×SD₁₆
Hall	C₃²×ΓL(1,𝔽₈) ⋊ SD₁₆
Sylow	C₇ ⋊ C₃×AΓL(1,𝔽₉)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₇×#157	C₂²×C₃
Fitting, socle, hypocenter	C₃²×C₇	C₃×SD₁₆
prime radical	#157⋊AGL(1,𝔽₇)	C₂
LCS	G ▷ C₇×#157 ▷ C₇×C₃⋊D₃ ▷ C₃²×C₇
UCS	C₁

108 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 108 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×AΓL(1,𝔽₉), C₂×#72224682
maximal normal subgroup of	AGL(1,𝔽₇)×AΓL(1,𝔽₉), C₂×#72224682
maximal quotient of	C₂×#72224682

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