C₂⁴×SD₁₆×C₅₃ #27270590

order	13568
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₅₃ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂⁴×SD₁₆×C₅₃

Subgroups

	subgroup	G/H
center, socle	C₂⁵×C₅₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₅₃
prime radical	C₂⁴×D₄×C₅₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

13310 conjugacy classes of subgroups; only the 6878 normal subgroups are shown.

normal subgroups only

This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 6878 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) = 6784

A minimal faithful permutation representation — 6 generators on 69 points:

(1,2)(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);(3,4);(5,6);(7,8);(9,10,12,15)(11,13,16,14);(10,13)(11,16)(14,15)

Conjugacy classes & character table

5936 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56066);