C₅₃×#13656 #27221168

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

Aliases

direct product

C₅₃×#13656

Subgroups

	subgroup	G/H
center	C₄×C₅₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	C₄×D₁₆
prime radical	C₅₃×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 184 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:

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

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6644);