C₄₁×#548 #14580016

order	2624
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄₁×#548

Subgroups

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ G

290 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 53
transitive degree	μ_t(G) = 1312

A minimal faithful permutation representation — 4 generators on 53 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)(47,50)(51,52);(46,47,49,50)(48,52,53,51);(42,43)(44,45);(46,48,49,53)(47,51,50,52)

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄₁×#3265, C₄₁×#3269, C₄₁×#3270, …
maximal normal subgroup of	C₄₁×#3265, C₄₁×#3269, C₄₁×#3270, …
maximal quotient of	Q₈×C₂²⋊C₄×C₄₁, C₄₁×#2341, C₄₁×#2351, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 229);