#72246256

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#41 ⋊ C₃×D₃
derived	C₇×SL(2,𝔽₃) ⋊ C₂²×C₃
Sylow	C₇ ⋊ C₃×#986

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×A₄⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂²×C₃
Fitting	C₇×#41	C₃×D₃
socle	C₂×C₇	C₂×C₃×S₄
hypercenter	C₄	A₄⋊AGL(1,𝔽₇)
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

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

normal subgroups only

This group has 178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72246258, AGL(1,𝔽₇)×#986, #72246274, …
maximal normal subgroup of	#72246258, AGL(1,𝔽₇)×#986, #72246274

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