C₅₃×#10689 #27218201

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#10689

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#400
prime radical	C₂³×C₅₃	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

248 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 points:

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

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, 3677);