C₅₃×#11859 #27219371

order	13568
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#11859

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

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

normal subgroups only

This group has 314 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) = 101
transitive degree	μ_t(G) = 3392

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

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,29)(20,34)(21,25)(22,40)(23,28)(24,31)(26,36)(27,38)(30,43)(32,48)(33,47)(35,44)(37,46)(39,45)(41,42)(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)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19,23,33)(18,24,28,44)(20,35,36,31)(21,37,39,30)(22,41,38,32)(25,29,45,47)(26,42,34,48)(27,46,40,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,21,23,36,38,39)(18,25,27,34,28,45,40,26)(19,30,32,31,33,37,41,35)(24,42,43,47,44,48,46,29)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4847);