C₂×C₅₃×#1308 #27215635

order	13568
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₅₃×#1308

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#95
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

526 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 87
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 87 points:

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

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, 1111);