C₅₃×#8919 #27216431

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#8919

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#385
prime radical	C₂³×C₅₃	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

336 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 3392

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

(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,22,19,21,24,30,25)(20,26,28,31,29,32,27,23)(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);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19,22,24,21,25,30,18)(20,26,28,31,29,32,27,23);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20)(18,23)(19,26)(21,29)(22,28)(24,31)(25,32)(27,30)

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, 1907);