C₅₃×#12759 #27220271

order	13568
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅₃×#12759

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×D₄	C₂³×C₅₃
prime radical	C₂×D₄²×C₅₃	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×D₄×C₅₃ ◁ G

1556 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 65
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 65 points:

(1,2,3,4)(5,6)(7,10,12,9)(8,11)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(6,9)(7,12)(10,11);(7,12)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5747);