C₅₃×#12593 #27220105

order	13568
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₅₃ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₅₃×#12593

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ G

338 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,3,5,4,6,7,8)(9,10,14,20,15,22,12,18,16,21,29,38,13,19,30,37)(11,23,26,33,27,17,32,34,28,36,31,40,25,35,24,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(9,11)(10,17,21,35)(12,24)(13,25)(14,31)(15,27)(16,28)(18,33,37,40)(19,23,22,36)(20,39,38,34)(26,29)(30,32);(9,12,13,14,16,30,15,29)(10,18,19,20,21,37,22,38)(11,24,25,31,28,32,27,26)(17,33,23,39,35,40,36,34)

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, 5581);