#72249124

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Hall	#13133672 ⋊ C₂
Hall	D₇ ⋊ 3₋¹⁺⁴
Sylow	C₇ ⋊ C₂×3₋¹⁺⁴

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₃×C₇	C₂×C₃⁴
Fitting	C₃×C₇×H(3,𝔽₃)	C₂×C₃
prime radical	C₃×D₇×H(3,𝔽₃)	C₃
hypercenter	C₃×H(3,𝔽₃)	AGL(1,𝔽₇)
hypocenter	C₇	C₂×3₋¹⁺⁴
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃)

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 189

Polycyclic presentation:

2,3^5,7|b^3=f,d^b=df,d^c=df,e^b=ef^2,g^a=g^6,g^b=g^2

Conjugacy classes & character table

199 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×3₋¹⁺⁴, #72249151, #72249152
maximal normal subgroup of	AGL(1,𝔽₇)×3₋¹⁺⁴, #72249151, #72249152

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