C₄₁×#554 #14580022

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

Aliases

direct product

C₄₁×#554

Subgroups

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₁
prime radical	C₂×D₄×C₄₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ G

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

normal subgroups only

This group has 228 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) = 57
transitive degree	μ_t(G) = 1312

A minimal faithful permutation representation — 4 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)(42,43,46,45)(44,48,49,47)(50,51,54,53)(52,56,57,55);(43,45)(47,48)(50,52,54,57)(51,55,53,56);(42,44)(43,47)(45,48)(46,49);(42,45,46,43)(44,48,49,47)(50,53,54,51)(52,55,57,56)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×#1622, C₄₁×#1632, C₄₁×#1643, …
maximal normal subgroup of	C₄₁×#1622, C₄₁×#1632, C₄₁×#1643, …
maximal quotient of	C₄₁×#2345, D₄×C₄⋊C₄×C₄₁, C₄₁×#2349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 235);