C₅₃×#13053 #27220565

order	13568
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13053

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

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

normal subgroups only

This group has 624 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)(3,12,15,26)(4,9)(5,10,20,8)(6,11)(7,13,24,16)(14,30)(17,25,21,27)(18,28)(19,29)(22,31)(23,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,16,6,15,18,13)(2,7,9,26,11,24,28,12)(5,19,17,31,20,14,21,32)(8,22,27,30,10,23,25,29);(2,8)(5,20)(7,22)(9,27)(10,11)(12,29)(14,19)(17,21)(23,24)(25,28)(26,30)(31,32)

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6041);