#27410662

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1902 ⋊ C₂
Sylow	C₅₃ ⋊ #23360

Subgroups

	subgroup	G/H
center	C₂	#20277890
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1902	C₂
socle	C₂×C₅₃	#2380
prime radical	#16811169	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1902	D₅₃
hypocenter	C₅₃	#23360
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1902

688 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)(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)(55,62,63,77)(60,75,64,79)(61,68,78,82)(74,85,80,76);(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)(54,57,58,70,59,71,73,84)(55,61,77,82,63,78,62,68)(56,65,66,69,67,81,72,83)(60,74,79,76,64,80,75,85);(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)(54,56)(55,60)(57,69)(58,72)(59,67)(61,76)(62,79)(63,64)(65,70)(66,73)(68,80)(71,83)(74,82)(75,77)(78,85)(81,84);(54,55)(56,64,67,60)(57,68,71,82)(58,62)(59,63)(61,70,78,84)(65,76)(66,79,72,75)(69,74)(73,77)(80,83)(81,85)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196138);