C₅₃×#7407 #27214919

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7407

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#353
commutator	C₂×M₁₆	C₂×C₄×C₅₃
Frattini, cosocle	#408	C₂²×C₅₃
socle	C₂×C₅₃	#1343
prime radical	C₅₃×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#408 ◁ G

240 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 117 points:

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

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 395);