C₄×#41×C₄₁ #14579985

order	2624
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₄₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#41×C₄₁

Subgroups

	subgroup	G/H
center	C₄²×C₄₁	C₂²
commutator	C₂	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂⁴
prime radical	C₂×#41×C₄₁	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₄₁ ◁ G

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 53 points:

(1,2,3,4)(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);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×#1476, C₄₁×#1477, C₄₁×#1488, …
maximal normal subgroup of	C₄₁×#1476, C₄₁×#1477, C₄₁×#1488, …
maximal quotient of	C₄×#117×C₄₁, C₄²×D₄×C₄₁, C₄²×Q₈×C₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 198);

C4×#41×C41 #14579985