C₅₃×#7057 #27214569

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7057

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#99
commutator	C₂×C₄	C₂×C₁₆×C₅₃
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#323
prime radical	C₂³×C₅₃	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄⋊C₈×C₅₃ ◁ G

196 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 6784

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)(17,18)(19,25,27,32)(20,28,30,23)(21,24)(22,26)(29,31)(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);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25)

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 45);