C₅₃×#12771 #27220283

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#12771

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1790
commutator	C₂×D₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

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

normal subgroups only

This group has 728 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:

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

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, 5759);