#72224656

order	2646
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Hall	(C₃×ΓL(1,𝔽₈))⋊ΓL(1,𝔽₈) ⋊ C₂
Hall	C₇⋊D₇ ⋊ H(3,𝔽₃)
Sylow	C₇² ⋊ C₂×H(3,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, Fitting, socle, cosocle	C₃×C₇²	C₂×C₃²
hypocenter	C₇²	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×C₇² ▷ C₇²
UCS	C₁ ◁ C₃

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 23 points:

(2,5,6)(4,8,9)(10,13,11,16,15,14)(18,21,22,19,20,23);(1,2,4)(3,6,8)(5,9,7)(11,12,16)(13,15,14)(17,18,20,23,22,21,19)

Polycyclic presentation:

2,3^3,7^2|c^b=cd,e^a=e^6,e^b=f,e^c=e^2,f^a=f^6,f^b=e^5f^3,f^c=f^2

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224660, #72230257, C₂×#72224656
maximal normal subgroup of	#72224660, #72230257, C₂×#72224656
maximal quotient of	C₂×#72224656

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