C₅₃×#13352 #27220864

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13352

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

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

normal subgroups only

This group has 682 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)(3,13,17,40)(4,18,21,37)(5,10,24,36)(6,25,27,11)(8,28,31,15)(9,26,33,20)(14,41,44,63)(16,46,48,39)(19,50,22,52)(23,55,56,35)(29,59,60,64)(30,45,57,58)(32,54,34,51)(38,47,62,43)(42,49,53,61)(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)(5,15)(6,16)(7,17)(9,29)(10,13)(11,30)(12,31)(18,49)(19,42)(20,43)(21,44)(22,53)(23,45)(24,28)(25,57)(26,47)(27,48)(32,38)(33,60)(34,62)(35,39)(36,40)(37,61)(41,50)(46,55)(51,64)(52,63)(54,59)(56,58);(1,4,7,21)(2,9,12,33)(3,14,17,44)(5,22,24,19)(6,26,27,20)(8,29,31,60)(10,34,36,32)(11,37,25,18)(13,38,40,62)(15,42,28,53)(16,47,48,43)(23,51,56,54)(30,61,57,49)(35,50,55,52)(39,63,46,41)(45,59,58,64)

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, 6340);