C₅₃×#13054 #27220566

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13054

Subgroups

	subgroup	G/H
center	C₄×C₅₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
socle	C₂×C₅₃	#1889
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,12,13,26)(4,9)(5,10,20,8)(6,11)(7,14,23,16)(15,29)(17,25,21,27)(18,28)(19,32)(22,31)(24,30)(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);(1,3,4,16,6,13,18,14)(2,7,9,26,11,23,28,12)(5,19,17,30,20,15,21,31)(8,24,27,29,10,22,25,32);(2,8)(3,13)(5,20)(7,22)(9,27)(10,11)(12,29)(14,16)(17,21)(23,24)(25,28)(26,32)

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, 6042);