C₅₃×#13684 #27221196

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13684

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₅₃
prime radical	C₅₃×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄⋊Q₈×C₅₃ ◁ G

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2,8,12)(3,13,18,17)(4,19,22,21)(5,11,6,7)(9,23,16,15)(10,24,26,20)(14,28,30,32)(25,27,29,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,8,18)(2,9,12,16)(4,14,22,30)(5,23,6,15)(7,17,11,13)(10,25,26,29)(19,31,21,27)(20,32,24,28);(1,4)(2,10)(3,14)(5,20)(6,24)(7,19)(8,22)(9,25)(11,21)(12,26)(13,27)(15,28)(16,29)(17,31)(18,30)(23,32)

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6672);