C₅₃×#7227 #27214739

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7227

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

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

normal subgroups only

This group has 248 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)(3,15,17,44)(4,12,22,39)(6,28,25,38)(7,13,26,40)(8,14,27,41)(9,30,31,48)(10,35,36,29)(16,34,47,37)(18,52,49,64)(19,32,50,59)(20,45,51,58)(21,42,54,62)(23,43,24,55)(33,57,60,53)(46,61,56,63)(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);(1,3,6,18,7,19,23,48,8,20,29,53,21,46,4,16)(2,9,12,32,42,52,10,15,14,34,43,61,13,33,28,45)(5,17,25,49,26,50,24,30,27,51,35,57,54,56,22,47)(11,37,39,63,62,60,36,58,41,31,55,59,40,64,38,44)

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, 215);