C₅₃×#12772 #27220284

order	13568
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#12772

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	#1790
prime radical	C₅₃×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

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, 5760);