C₅₃×#7142 #27214654

order	13568
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₅₃ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₅₃×#7142

Subgroups

	subgroup	G/H
center	C₄²×C₅₃	C₄²
commutator	C₄	C₄×C₁₆×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ G

246 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 3392

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

Conjugacy classes & character table

4664 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 130);