#72259789

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×Q₈ ⋊ C₃×D₃
Sylow	C₇ ⋊ C₃×#983

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×A₄⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂²×C₃
Fitting	C₂×C₇×Q₈	C₃×D₃
socle	C₂×C₇	C₂×C₃×S₄
hypercenter	C₂²	A₄⋊AGL(1,𝔽₇)
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂²

170 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 23
transitive degree	μ_t(G) = 112

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

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

Polycyclic presentation:

2^2,3^2,7,2^3|f^2=h,g^2=h,b^a=bh,d^b=d^2,e^b=e^6,e^c=e^2,f^b=gh,f^d=fg,g^b=fh,g^d=f,g^f=gh

Conjugacy classes & character table

53 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 7, 2, 2, 2]);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^6);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 4, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);