C₄₁×#464 #14579932

order	2624
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄₁×#464

Subgroups

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

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 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)(43,45)(46,47)(48,54)(49,51)(50,58)(52,56)(53,57)(55,61)(59,60);(42,43,44,45)(47,51)(52,60)(53,58)(54,61);(46,48,49,55)(47,52,51,60)(50,59,57,56)(53,61,58,54)

Conjugacy classes & character table

779 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄₁×#464, C₄₁×#3035, C₄₁×#3037, …
maximal normal subgroup of	C₂×C₄₁×#464, C₄₁×#3035, C₄₁×#3037, …
maximal quotient of	C₄₁×#1678, C₄₁×#1680, C₄₁×#1682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 145);