C₄×Q₁₆×C₄₁ #14579907

order	2624
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₄₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×Q₁₆×C₄₁

Subgroups

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

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 2624

A minimal faithful permutation representation — 3 generators on 61 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(5,6,7,8,9,10,11,12)(13,20,19,18,17,16,15,14);(5,13,9,17)(6,14,10,18)(7,15,11,19)(8,16,12,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×#1332, C₄₁×#1334, C₈×Q₁₆×C₄₁, …
maximal normal subgroup of	C₄₁×#1332, C₄₁×#1334, C₈×Q₁₆×C₄₁, …
maximal quotient of	C₈×Q₁₆×C₄₁, C₄₁×#1584, C₄₁×#1591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 120);

C4×Q16×C41 #14579907