C₄₁×#421 #14579889

order	2624
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₄₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₄₁×#421

Subgroups

	subgroup	G/H
center	C₄×C₄₁	C₂×D₄
commutator	C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄	C₂³×C₄₁
socle	C₂×C₄₁	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₄₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 656

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

Conjugacy classes & character table

902 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄₁×#1794, C₄₁×#1795, C₄₁×#1796, …
maximal normal subgroup of	C₄₁×#1794, C₄₁×#1795, C₄₁×#1796, …
maximal quotient of	C₄₁×#1477, C₄₁×#1478, C₄₁×#1479, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 102);