C₅₃×#7071 #27214583

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7071

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#102
commutator	C₈	C₂×C₁₆×C₅₃
Frattini, cosocle	C₈²	C₂²×C₅₃
socle, prime radical	C₂²×C₅₃	D₄⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₈²×C₅₃ ◁ G

140 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 13568

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

3392 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59);