#72242276

order	7488
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃ ⋊ C₃×(C₂×S₄)⋊C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×AGL(1,𝔽₁₃)
commutator	C₂×A₄×C₁₃	C₂×C₃×C₄
Fitting, socle	C₂³×C₁₃	C₃×C₄×D₃
prime radical	C₂×S₄×D₁₃⋊C₃	C₂
cosocle	C₂×A₄×D₁₃	C₂²×C₃
hypocenter	A₄×C₁₃	C₃×C₂²⋊C₄
LCS	G ▷ C₂×A₄×C₁₃ ▷ A₄×C₁₃
UCS	C₁ ◁ C₂

604 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 21
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2^2,3,2^2,3,13,2^2|b^2=d,b^a=be,f^a=f^2,g^b=g^8,g^c=g^3,g^d=g^12,h^f=i,i^a=hi,i^f=hi

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	GL(2,ℤ/4)×AGL(1,𝔽₁₃)
maximal normal subgroup of	GL(2,ℤ/4)×AGL(1,𝔽₁₃)

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, 3, 2, 2, 3, 13, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);