C₄₁×#547 #14580015

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

Aliases

direct product

C₄₁×#547

Subgroups

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

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

normal subgroups only

This group has 292 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)(46,47,50,51)(48,49,52,53);(47,51)(49,53);(42,43)(44,45)(46,48,50,52)(47,49,51,53);(46,49)(47,52)(48,51)(50,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×#1625, C₄₁×#1635, C₄₁×#1647, …
maximal normal subgroup of	C₄₁×#1625, C₄₁×#1635, C₄₁×#1647, …
maximal quotient of	C₄₁×#2339, C₄₁×#2344, D₄×C₄⋊C₄×C₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 228);