D₅₃×#1416 #27304519

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₅₃×#1416

Splittings

Fitting	C₅₃×#1416 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1416

Subgroups

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1416	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂²×D₅₃	D₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1416	D₅₃
hypocenter	C₅₃	C₂×#1416
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

302 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 89 points:

(5,6,10,14)(7,16,19,33)(8,20,22,36)(9,23,21,13)(11,24,26,15)(12,28,29,17)(18,35,31,27)(25,32,30,34)(37,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39);(1,2,3,4)(6,11)(7,15)(8,17)(9,21)(12,27)(13,30)(14,26)(16,20)(18,34)(19,24)(22,28)(23,25)(29,35)(31,32)(33,36);(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)(86,87)(88,89)

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89995);