#72245017

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×AGL(2,𝔽₃)
commutator	C₂×PSU(3,𝔽₂)⋊C₃	C₂×C₄
Fitting	C₂×C₃²×C₄	GL(2,𝔽₃)
socle	C₂²×C₃²	C₂×GL(2,𝔽₃)
prime radical, cosocle	C₂²×PSU(3,𝔽₂)⋊C₃	C₂²
hypercenter	C₂×C₄	AGL(2,𝔽₃)
hypocenter	PSU(3,𝔽₂)⋊C₃	C₄⋊C₄
LCS	G ▷ C₂×PSU(3,𝔽₂)⋊C₃ ▷ PSU(3,𝔽₂)⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

468 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 17 points:

(1,6,8,4,3,2)(5,7)(14,15,16,17);(2,3)(4,9)(5,8)(10,11,12,13)(15,17)

Polycyclic presentation:

2^4,3,2^3,3^2|a^2=c,b^2=d,f^2=h,g^2=h,b^a=bd,e^a=e^2,f^a=fh,f^e=g,g^a=fg,g^e=fgh,g^f=gh,i^a=j,i^e=j,i^f=j,i^g=ij^2,i^h=i^2,j^a=i,j^e=i^2j^2,j^f=i^2,j^g=i^2j^2,j^h=j^2

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245046, C₄⋊C₄×AGL(2,𝔽₃), #72269011
maximal normal subgroup of	#72245046, C₄⋊C₄×AGL(2,𝔽₃), #72269011
maximal quotient of	#72245046, #72245256

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 8, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 5, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 6, F.9^2);;
gap> SetConjugate(coll, 10, 7, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 8, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);