C₂×C₅₃×#1316 #27218935

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₅₃×#1316

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#99
commutator	C₂³	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅₃
prime radical	C₂⁶×C₅₃	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂⁴×C₅₃ ◁ C₂²×C₂²⋊C₄×C₅₃ ◁ G

1886 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 71 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);(56,57,58,60,59,61,62,63)(64,65)(66,70,71,68)(67,69);(65,68)(69,70)

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, 4411);