#72229136

order	2496
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₂⁴×C₁₃ ⋊ C₃×C₄
Hall	C₂⁴×#261 ⋊ C₃
Sylow	C₁₃ ⋊ C₄×C₂²⋊A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₂⁴×C₁₃	C₃×C₄
prime radical	D₁₃⋊(C₂²⋊A₄)	C₂
cosocle	C₂⁴×D₁₃	C₂×C₃
LCS	G ▷ C₂⁴×C₁₃
UCS	C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

2,3,2^5,13|a^2=c,d^b=de,e^b=d,f^b=fg,g^b=f,h^a=h^8,h^b=h^3,h^c=h^12

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊A₄×AGL(1,𝔽₁₃), #72242039, A₄×#261⋊A₄
maximal normal subgroup of	C₂²⋊A₄×AGL(1,𝔽₁₃), #72242039, A₄×#261⋊A₄
maximal quotient of	#72229138

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