C₁₃×#6122 #72229263

order	2704
isoclinism	isoclinic class of #6118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₃×#6122

Splittings

Sylow

C₁₃² ⋊ M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₁₃	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₃
Frattini	C₂	C₂×C₁₃×#261
Fitting, prime radical	C₂²×C₁₃²	C₄
socle	C₂×C₁₃²	C₂×C₄
cosocle	Dic₁₃	C₂²×C₁₃
hypercenter	C₂²×C₁₃	#261
hypocenter	C₁₃	C₁₃×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₁₃

55 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,21,25,19,23,24,16,20,26,22,17)(28,31)(32,34);(15,16,17,19)(18,21,22,26)(20,24,25,23)(27,28,29,32,30,31,33,34)

Conjugacy classes & character table

286 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72229265, #72229278
maximal normal subgroup of	#72229265, #72229278

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,21,25,19,23,24,16,20,26,22,17)(28,31)(32,34), (15,16,17,19)(18,21,22,26)(20,24,25,23)(27,28,29,32,30,31,33,34));

C13×#6122 #72229263