C₄₁×#452 #14579920

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

Aliases

direct product

C₄₁×#452

Subgroups

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

152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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)(42,43)(44,45)(46,47,51,54)(48,55,57,60)(49,53,50,58)(52,59,61,56);(46,48)(47,52)(49,56)(50,59)(51,57)(53,60)(54,61)(55,58);(43,45)(46,49,51,50)(47,53,54,58)(48,56,57,59)(52,60,61,55)

Conjugacy classes & character table

779 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄₁×#452, C₄₁×#3002, C₄₁×#3005, …
maximal normal subgroup of	C₂×C₄₁×#452, C₄₁×#3002, C₄₁×#3005, …
maximal quotient of	C₄₁×#1603, C₄₁×#1606, C₄₁×#1615, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 133);