C₅₃×#13356 #27220868

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13356

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅₃
prime radical	C₅₃×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

676 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6344);