C₅₃×#13431 #27220943

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13431

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅₃
prime radical	C₂×#41×C₅₃	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

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

normal subgroups only

This group has 402 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 — 3 generators on 117 points:

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

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6419);