C₅₃×#13709 #27221221

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#13709

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₅₃×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

154 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6697);