C₅₃×#11484 #27218996

order	13568
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#11484

Subgroups

	subgroup	G/H
center	C₄×C₅₃	C₂×#99
commutator	C₂³	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
socle	C₂×C₅₃	#1459
prime radical	C₅₃×#525	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

368 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 1696

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

(1,2,4,9,5,10,15,22,6,7,14,18,16,19,26,29)(3,8,12,20,13,21,25,30,11,17,23,27,24,28,31,32)(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);(2,7)(3,11)(10,19)(12,23)(13,24)(20,27)(25,31)(30,32);(1,3,5,13,6,11,16,24)(2,8,10,21,7,17,19,28)(4,12,15,25,14,23,26,31)(9,20,22,30,18,27,29,32)

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4472);