C₄₁×#339 #14579807

order	2624
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄²×C₄₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄₁×#339

Subgroups

	subgroup	G/H
center	C₂×C₄×C₄₁	D₄
commutator	C₄	C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂²×C₄₁
socle	C₂²×C₄₁	C₂²⋊C₄
prime radical	C₂³×C₄₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 656

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)(42,43,44,45)(46,47)(48,50)(49,52)(51,53);(47,50,52,51)

Conjugacy classes & character table

1148 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄₁×#339, C₄₁×#1733, C₄×#104×C₄₁, …
maximal normal subgroup of	C₂×C₄₁×#339, C₄₁×#1733, C₄×#104×C₄₁, …
maximal quotient of	C₄₁×#1276, C₄₁×#1278, C₄₁×#1279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 20);

C41×#339 #14579807