D₅₃×#1356 #27295264

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

Aliases

direct product

D₅₃×#1356

Splittings

Fitting	C₅₃×#1356 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1356

Subgroups

	subgroup	G/H
center	C₂	#329×D₅₃
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#97	C₂²×D₅₃
Fitting	C₅₃×#1356	C₂
socle	C₂×C₅₃	C₂×#329
prime radical	C₂²×D₅₃	#100
cosocle	#97×C₅₃	C₂³
hypercenter	#1356	D₅₃
hypocenter	C₅₃	C₂×#1356
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #1356

314 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 71 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,59,68)(55,60,64,77)(57,69,70,74)(58,72,73,67)(61,76,63,65)(62,80,82,83)(66,71,84,78)(75,79,85,81);(54,55,57,62,58,63,71,81,59,64,70,82,73,61,78,79)(56,65,66,83,67,60,74,75,68,76,84,80,72,77,69,85);(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, 80740);