D₅₃×#1409 #27304398

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

Aliases

direct product

D₅₃×#1409

Splittings

Fitting	C₅₃×#1409 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1409

Subgroups

	subgroup	G/H
center	C₂	#353×D₅₃
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#127	C₂²×D₅₃
Fitting	C₅₃×#1409	C₂
socle	C₂×C₅₃	C₂×#353
prime radical	#127×D₅₃	C₂²
cosocle	#127×C₅₃	C₂³
hypercenter	#1409	D₅₃
hypocenter	C₅₃	C₂×#1409
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #1409

474 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 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)(54,56,58,65)(55,59,62,69)(57,66,67,64)(60,68,61,63);(54,55)(56,63,64,59)(57,61,67,60)(58,62)(65,68,66,69);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89874);