D₅₃×#1340 #27294235

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

Aliases

direct product

D₅₃×#1340

Splittings

Fitting	C₅₃×#1340 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1340

Subgroups

	subgroup	G/H
center	C₂	#327×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1340	C₂
socle	C₂×C₅₃	C₂×#327
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1340	D₅₃
hypocenter	C₅₃	C₂×#1340
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1340

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

normal subgroups only

This group has 444 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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79711);