#72259779

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×H(5,𝔽₂) ⋊ C₂×C₃
Hall	#11741549 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3019
Sylow	C₇ ⋊ C₃×#3019

Subgroups

	subgroup	G/H
center	C₂	C₂×#10620789
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₂×C₇×H(5,𝔽₂)	C₂×C₃
socle	C₂×C₇	C₂×C₃×#120
hypercenter	C₂×H(5,𝔽₂)	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3019
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×H(5,𝔽₂)

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^4,3,2^3,7|c^2=h,f^2=h,b^a=bh,c^b=cf,d^b=dg,d^c=dh,f^b=fh,f^c=fh,f^d=fh,g^c=gh,i^b=i^6,i^e=i^2

Conjugacy classes & character table

121 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72259795

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 7]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);