C₅₃×#10384 #27217896

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#10384

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂×#121×C₅₃	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

438 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,5)(4,7)(6,8)(9,10)(11,14)(12,19,21,15)(13,16)(17,22,20,24)(18,23)(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);(3,8)(5,6)(9,11,13,18)(10,14,16,23)(12,20)(15,24)(17,21)(19,22);(2,5)(3,8)(6,7)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24)

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 3372);