C₅₃×#7395 #27214907

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7394
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7395

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1343
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₅₃
Frattini, cosocle	#381	C₂²×C₅₃
prime radical	C₅₃×#534	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂³×C₅₃ ◁ C₂⁴×C₅₃ ◁ C₅₃×#381 ◁ G

284 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 85 points:

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

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 383);