C₅₃×#12284 #27219796

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12284

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₅₃×#3199	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#408 ◁ G

338 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18,22,27)(19,28,32,42)(20,23,36,40)(21,37,35,26)(24,31,43,30)(25,44,39,33)(29,41,34,45)(38,47,48,46)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(18,23)(20,33)(21,35)(24,38)(25,26)(27,40)(28,41)(29,46)(30,31)(34,47)(36,44)(37,39)(42,45)(43,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,24)(20,34)(21,31)(22,32)(23,38)(25,45)(26,42)(27,43)(28,37)(29,36)(30,35)(33,47)(39,41)(40,48)(44,46)

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, 5272);