C₅₃×#13435 #27220947

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13435

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

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

normal subgroups only

This group has 408 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)(3,13,17,37)(4,18,22,49)(5,23,26,10)(6,27,29,11)(8,24,32,15)(9,33,36,21)(14,42,19,50)(16,46,28,41)(20,53,55,48)(25,38,59,57)(30,60,34,63)(31,45,39,58)(35,54,56,62)(40,52,64,44)(43,61,51,47)(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)(6,28,29,16)(9,34,36,30)(10,37,23,13)(11,39,27,31)(18,47,49,61)(20,43,55,51)(21,52,33,44)(25,45,59,58)(35,40,56,64)(38,41,57,46)(42,53,50,48)(54,63,62,60);(1,4,5,20,7,22,26,55)(2,9,10,35,12,36,23,56)(3,14,15,43,17,19,24,51)(6,21,25,54,29,33,59,62)(8,30,13,40,32,34,37,64)(11,18,38,48,27,49,57,53)(16,44,45,63,28,52,58,60)(31,61,41,42,39,47,46,50)

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, 6423);