C₅₃×#12292 #27219804

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12292

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄⋊C₈×C₅₃ ◁ G

310 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

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

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, 5280);