#27353055

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1560 ⋊ C₂
Sylow	C₅₃ ⋊ #17782

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1560	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#20288488	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1560	D₅₃
hypocenter	C₅₃	#17782
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1560

1244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 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)(62,66,68,64)(63,71,72,73)(65,74,67,75)(69,76,77,70);(62,65,68,67)(63,70,72,76)(64,74,66,75)(69,71,77,73);(55,58)(56,60)(62,64,68,66)(63,69,72,77)(65,74,67,75)(70,71,76,73);(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)(54,55,56,59,57,58,60,61)(62,63)(64,73)(65,70)(66,71)(67,76)(68,72)(69,74)(75,77)

Polycyclic presentation:

2^8,53|a^2=f,b^2=h,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=be,e^a=eg,i^a=i^52,e^b=eh,f^b=fgh,d^c=dh,i^d=i^52

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 138531);