C₅₃×#12932 #27220444

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#12932

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#481
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1853
prime radical	C₂⁴×C₅₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

402 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2,5,9,6,11,21,10)(3,12,15,30,14,29,20,26)(4,16,19,8,18,28,31,27)(7,22,24,13,23,17,25,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,6,14)(2,7)(4,17,18,22)(5,20,21,15)(8,26)(9,25)(10,24)(11,23)(12,16)(13,31,32,19)(27,30)(28,29);(4,18)(8,27)(12,29)(13,32)(16,28)(17,22)(19,31)(26,30)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5920);