D₅₃×#1419 #27304557

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

Aliases

direct product

D₅₃×#1419

Splittings

Fitting	C₅₃×#1419 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1419

Subgroups

	subgroup	G/H
center	C₂	#357×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1419	C₂
socle	C₂×C₅₃	C₂×#357
prime radical	C₂×D₁₆×D₅₃	C₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1419	D₅₃
hypocenter	C₅₃	C₂×#1419
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1419

326 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 81 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 — 3 generators on 85 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90033);