C₅₃×#11311 #27218823

order	13568
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#11311

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#396
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂×C₅₃	#1443
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

532 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 202 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)(3,13)(4,17,21,9)(5,11)(6,12)(7,10)(8,15)(14,25,29,30)(16,28)(18,23,31,32)(19,26,20,27)(22,24)(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);(1,3,7,15,6,16,5,22)(2,8,10,28,12,24,11,13)(4,18,19,14,21,31,20,29)(9,25,27,32,17,30,26,23);(1,4,5,19,6,21,7,20)(2,9,10,26,12,17,11,27)(3,14,15,18,16,29,22,31)(8,23,13,25,24,32,28,30)

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4299);