C₅₃×#13647 #27221159

order	13568
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13647

Subgroups

	subgroup	G/H
center	C₃₂×C₅₃	D₄
commutator	C₄	C₂²×C₁₆×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	D₄×C₁₆
prime radical	#135×C₅₃	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₅₃ ◁ C₂×C₃₂×C₅₃ ◁ G

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Conjugacy classes & character table

5936 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6635);