C₅₃×#13051 #27220563

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13051

Subgroups

	subgroup	G/H
center	C₄×C₅₃	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
socle	C₂×C₅₃	#1889
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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:

(1,2,7,10,8,13,22,28)(3,14,19,29,20,30,31,26)(4,21,6,12,5,11,15,24)(9,18,25,16,27,23,32,17)(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);(1,3,15,18,8,20,6,23)(2,9,24,29,13,27,12,26)(4,16,22,31,5,17,7,19)(10,25,21,30,28,32,11,14);(1,4)(2,10)(3,16)(5,8)(6,7)(9,25)(11,12)(13,28)(14,26)(15,22)(17,20)(18,31)(19,23)(21,24)(27,32)(29,30)

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, 6039);