C₅₃×#9336 #27216848

order	13568
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#9336

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

360 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 6784

A minimal faithful permutation representation — 3 generators on 117 points:

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

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, 2324);