C₅₃×#7215 #27214727

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#7215

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#342
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₂²×C₄⋊C₄	C₂²×C₅₃
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄⋊C₄×C₅₃ ◁ G

422 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 69 points:

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

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, 203);