C₅₃×#12724 #27220236

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×#12724

Subgroups

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

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

normal subgroups only

This group has 670 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)(4,10,20,9)(5,11,19,28)(7,13,23,27)(8,21,25,15)(16,30,32,26)(18,31,22,29)(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)(4,15,19,32)(5,21,20,16)(6,22,23,17)(9,24,11,29)(10,14,28,31)(12,25,27,30);(2,9,13,11)(4,19)(6,23)(8,24,26,29)(10,12,28,27)(14,25,31,30)(15,32)(17,22)

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, 5712);