C₈×D₄×C₄₁ #14579902

order	2624
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₄₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈×D₄×C₄₁

Subgroups

	subgroup	G/H
center	C₂×C₈×C₄₁	C₂²
commutator	C₂	C₂²×C₈×C₄₁
Frattini, cosocle	C₂×C₄	C₂³×C₄₁
socle	C₂²×C₄₁	C₂²×C₄
prime radical	C₂×D₄×C₄₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₁ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 1312

A minimal faithful permutation representation — 3 generators on 53 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×D₄⋊C₁₆, C₈×D₈×C₄₁, C₈×SD₁₆×C₄₁, …
maximal normal subgroup of	C₄₁×D₄⋊C₁₆, C₈×D₈×C₄₁, C₈×SD₁₆×C₄₁, …
maximal quotient of	C₈×D₈×C₄₁, C₈×SD₁₆×C₄₁, C₈×Q₁₆×C₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 115);

C8×D4×C41 #14579902