C₅₃×#13691 #27221203

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13691

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₅₃×#2263	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 200 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)(3,15)(4,19)(5,25)(6,28)(7,9)(8,14)(10,21)(11,27)(12,30)(13,18)(16,46)(17,40)(20,52)(22,56)(23,51)(24,53)(26,58)(29,43)(31,45)(32,41)(33,36)(34,61)(35,60)(37,49)(38,54)(39,55)(42,62)(44,47)(48,64)(50,57)(59,63)(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);(1,3,5,7,8,18,27,33)(2,9,11,15,14,36,25,13)(4,16,20,32,24,22,23,48)(6,17,26,30,31,50,60,62)(10,34,37,44,39,63,38,43)(12,35,40,45,42,58,57,28)(19,41,51,46,53,64,52,56)(21,47,54,61,55,29,49,59);(1,4)(2,10)(3,16)(5,20)(6,29)(7,32)(8,24)(9,34)(11,37)(12,41)(13,43)(14,39)(15,44)(17,49)(18,22)(19,28)(21,30)(23,27)(25,38)(26,59)(31,47)(33,48)(35,51)(36,63)(40,46)(42,64)(45,53)(50,54)(52,58)(55,62)(56,57)(60,61)

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, 6679);