C₅₃×#7401 #27214913

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7401

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1343
commutator	C₂²×C₈	C₂×C₄×C₅₃
Frattini, cosocle	#408	C₂²×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#408 ◁ G

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

normal subgroups only

This group has 234 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) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2,5,10,7,12,21,37)(3,13,15,44,18,45,16,52)(4,19,22,55,24,56,47,63)(6,27,28,42,29,11,26,43)(8,30,31,58,34,20,32,57)(9,35,38,23,40,59,39,60)(14,46,48,41,50,62,49,33)(17,36,51,54,25,61,53,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,7,18)(2,8,12,34)(4,20,24,30)(5,25,21,17)(6,16,29,15)(9,36,40,61)(10,41,37,33)(11,32,27,31)(13,38,45,39)(14,47,50,22)(19,54,56,64)(23,58,60,57)(26,51,28,53)(35,49,59,48)(42,46,43,62)(44,63,52,55)

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, 389);