#27421609

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3585 ⋊ C₂
Sylow	C₅₃ ⋊ #24420

Subgroups

	subgroup	G/H
center	C₂	#20278089
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3585	C₂
socle	C₂×C₅₃	#2410
hypercenter	#3585	D₅₃
hypocenter	C₅₃	#24420
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3585

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 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)(55,63)(56,68)(58,74)(59,72)(60,73)(61,71)(62,76)(64,79)(65,78)(66,80)(67,81)(69,82)(70,83)(75,84)(77,85);(54,56)(55,62)(57,68)(58,67)(59,70)(60,69)(61,75)(63,76)(64,77)(65,66)(71,84)(72,83)(73,82)(74,81)(78,80)(79,85);(55,61)(56,67)(58,59)(62,66)(63,71)(68,81)(69,70)(72,74)(75,77)(76,80)(82,83)(84,85);(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)(54,55,58,64,60,61,59,65)(56,66,70,75,69,77,67,62)(57,71,72,79,73,63,74,78)(68,80,81,76,82,85,83,84)

Polycyclic presentation:

2^8,53|a^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,i^a=i^52,c^b=cg,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,f^d=fh

Conjugacy classes & character table

1106 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 207085);