C₅₃×#13693 #27221205

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

Aliases

direct product

C₅₃×#13693

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2206
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

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

normal subgroups only

This group has 196 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:

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

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, 6681);