C₅₃×#7431 #27214943

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7431

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₄×C₅₃
Frattini, cosocle	#391	C₂²×C₅₃
prime radical	C₂³×C₅₃	#99
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#391 ◁ G

206 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 77 points:

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

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 419);