C₅₃×#13646 #27221158

order	13568
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13646

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₅₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂×D₄×C₅₃	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 202 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2)(3,4)(5,6,9,14,10,15,23,29,11,16,13,27,24,30,28,36,8,12,20,25,21,26,34,35,22,7,17,18,31,19,32,33)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,4)(6,12)(7,16)(14,25)(15,26)(18,27)(19,30)(29,35)(33,36);(5,7,9,18,10,19,23,33,11,6,13,14,24,15,28,29,8,16,20,27,21,30,34,36,22,12,17,25,31,26,32,35)

Conjugacy classes & character table

5936 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6634);