#27304402

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2126 ⋊ C₂
Sylow	C₅₃ ⋊ #13583

Subgroups

	subgroup	G/H
center	C₄	#16810415
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#127	C₂²×D₅₃
Fitting	C₅₃×#2126	C₂
socle	C₂×C₅₃	C₂×#353
prime radical	#20285141	C₂
cosocle	#127×C₅₃	C₂³
hypercenter	#2126	D₅₃
hypocenter	C₅₃	#13583
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #2126

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

normal subgroups only

This group has 416 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)(54,57,59,70)(55,61,64,78)(56,66,68,81)(58,69,72,83)(60,73,75,85)(62,76,63,77)(65,79,74,84)(67,80,71,82);(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)(54,56)(55,60)(57,66)(58,71)(59,68)(61,73)(62,74)(63,65)(64,75)(67,72)(69,82)(70,81)(76,84)(77,79)(78,85)(80,83);(54,55)(56,65,67,60)(57,61)(58,63,72,62)(59,64)(66,79,80,73)(68,74,71,75)(69,77,83,76)(70,78)(81,84,82,85)

Polycyclic presentation:

2^8,53|a^2=e,c^2=h,d^2=gh,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dgh,e^b=efg,f^b=fh,i^b=i^52,e^d=egh,f^e=fh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89878);