C₄₁×#405 #14579873

order	2624
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₄₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄₁×#405

Subgroups

	subgroup	G/H
center	C₂×C₈×C₄₁	C₂²
commutator	C₂	C₂×C₄²×C₄₁
Frattini, cosocle	C₂×C₄	C₂³×C₄₁
socle	C₂²×C₄₁	C₂²×C₄
prime radical	C₂³×C₄₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₁ ◁ G

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(46,47,49,52,50,53,56,59)(48,51,55,58,54,57,60,61);(42,43,44,45)(46,48)(47,51)(49,55)(50,54)(52,58)(53,57)(56,60)(59,61);(48,54)(51,57)(55,60)(58,61)

Conjugacy classes & character table

1640 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄₁×#1378, C₄₁×#1379, C₄₁×#1747, …
maximal normal subgroup of	C₄₁×#1378, C₄₁×#1379, C₄₁×#1747, …
maximal quotient of	C₈×M₁₆×C₄₁, C₄₁×#1450, C₄₁×#1451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 86);

C41×#405 #14579873