C₅₃×#7491 #27215003

order	13568
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7491

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#99
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄⋊Q₈×C₅₃ ◁ G

276 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 73 points:

(5,6,8,12,10,7,14,16)(9,13,19,11,20,17,18,15)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,2,3,4)(5,7)(6,10)(8,11)(9,13)(12,19)(14,15)(16,18)(17,20)

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 479);