C₅₃×#12723 #27220235

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#12723

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1782
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
prime radical	C₅₃×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

694 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

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, 5711);