#27439690

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3584 ⋊ C₂
Sylow	C₅₃ ⋊ #25799

Subgroups

	subgroup	G/H
center	C₂	#20278348
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3584	C₂
socle	C₂×C₅₃	#2433
hypercenter	#3584	D₅₃
hypocenter	C₅₃	#25799
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3584

1458 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)(54,57,70,83,59,72,58,71)(55,61,73,82,63,75,62,69)(56,66,78,85,68,81,67,80)(60,74,77,84,65,79,64,76);(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)(56,65,68,60)(57,61)(58,62)(59,63)(64,67,77,78)(66,79,81,74)(69,71)(70,73)(72,75)(76,80,84,85)(82,83);(54,56,58,67,59,68,70,78)(55,60,62,64,63,65,73,77)(57,66,71,80,72,81,83,85)(61,74,69,76,75,79,82,84);(54,55)(56,64)(57,69)(58,73)(59,63)(60,67)(61,71)(62,70)(65,78)(66,79)(68,77)(72,82)(74,81)(75,83)(76,80)(84,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 225166);