C₅₃×#12046 #27219558

order	13568
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12046

Subgroups

	subgroup	G/H
center	C₁₆×C₅₃	D₈
commutator	C₈	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂×C₅₃	C₈×D₈
prime radical	C₅₃×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

216 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

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, 5034);