D₅₃×#1662 #27365252

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#1662

Splittings

Fitting	C₅₃×#1662 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1662

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1662	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#562×D₅₃	C₂
hypercenter	#1662	D₅₃
hypocenter	C₅₃	C₂×#1662
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1662

958 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 3392

A minimal faithful permutation representation — 4 generators on 85 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150728);