D₅₃×#1418 #27304549

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

Aliases

direct product

D₅₃×#1418

Splittings

Fitting	C₅₃×#1418 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1418

Subgroups

	subgroup	G/H
center	C₂	#357×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1418	C₂
socle	C₂×C₅₃	C₂×#357
prime radical	#508×D₅₃	C₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1418	D₅₃
hypocenter	C₅₃	C₂×#1418
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

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

normal subgroups only

This group has 302 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:

(1,2)(3,12)(4,17)(5,20)(6,8)(7,21)(9,16)(10,18)(11,23)(13,26)(14,27)(15,24)(19,22)(25,30)(28,32)(29,31)(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);(2,7,8,23)(3,11)(4,16)(5,18)(9,27,26,31)(10,28,20,25)(12,17,24,22)(13,19)(14,32)(15,21)(29,30);(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, 90025);