C₅₃×#7228 #27214740

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#7228

Subgroups

	subgroup	G/H
center	C₄×C₅₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×#131	C₂²×C₅₃
socle	C₂×C₅₃	#1289
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ G

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2,5,11)(3,15,17,43)(4,21,23,52)(6,28,25,54)(7,13,26,40)(8,14,27,41)(9,30,31,18)(10,35,36,29)(12,42,39,24)(16,34,47,38)(19,45,49,63)(20,32,50,58)(22,37,53,60)(33,57,59,51)(44,62,61,48)(46,55,56,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,6,18,22,46,24,48,8,20,4,16,7,19,29,51)(2,9,12,32,13,33,28,55,14,34,10,15,37,44,21,45)(5,17,25,30,53,56,42,62,27,50,23,47,26,49,35,57)(11,38,39,43,40,61,54,63,41,31,36,58,60,59,52,64)

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, 216);