C₅₃×#7092 #27214604

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7092

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#331
commutator	C₄×C₈	C₂×C₄×C₅₃
Frattini, cosocle	C₈²	C₂²×C₅₃
prime radical	C₅₃×#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

152 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 6784

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

2279 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80);