C₄₁×#341 #14579809

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

Aliases

direct product

C₄₁×#341

Subgroups

	subgroup	G/H
center	C₂×C₄×C₄₁	D₄
commutator	C₄	C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂²×C₄₁
socle	C₂²×C₄₁	C₄⋊C₄
prime radical	C₂³×C₄₁	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1312

A minimal faithful permutation representation — 2 generators on 61 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)(46,47,49,54,51,55,50,53)(48,56,57,52,59,60,58,61);(43,45)(46,48,50,58,51,59,49,57)(47,52,54,60,55,61,53,56)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×#1349, C₄₁×#1350, C₂×C₄₁×#341, …
maximal normal subgroup of	C₄₁×#1349, C₄₁×#1350, C₂×C₄₁×#341, …
maximal quotient of	C₄₁×#1277, C₄₁×#1278, C₄₁×#1287, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 22);

C41×#341 #14579809