C₅₃×#13657 #27221169

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13657

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2177
commutator	C₁₆	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₅₃×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₅₃×#346 ◁ G

182 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,15,18,48)(4,21)(5,24)(6,10)(7,33)(8,14)(9,17,36,55)(11,27)(12,23)(13,31)(16,51,19,57)(20,58,34,50)(22,42)(25,44,54,47)(26,61)(28,41)(29,46)(30,49,52,40)(32,39)(35,63,64,62)(37,59,45,56)(38,60)(43,53)(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)(2,9)(4,16)(5,25)(6,30)(7,34)(8,18)(10,35)(11,40)(12,44)(13,45)(14,36)(15,46)(17,53)(19,23)(20,31)(21,47)(22,59)(24,48)(26,56)(27,55)(28,63)(29,54)(32,52)(33,37)(38,50)(39,64)(41,58)(42,57)(43,49)(51,61)(60,62);(1,4,7,6,8,23,31,32)(2,10,13,21,14,39,33,12)(3,16,34,30,18,19,20,52)(5,26,28,53,29,22,60,27)(9,35,45,47,36,64,37,44)(11,41,42,24,43,38,61,46)(15,40,58,57,48,49,50,51)(17,54,59,62,55,25,56,63)

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6645);