#27387940

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1973 ⋊ C₂
Sylow	C₅₃ ⋊ #21228

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1973	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×#13133238	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1973	D₅₃
hypocenter	C₅₃	#21228
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1973

1582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

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)(62,66,68,67)(63,71,72,76)(64,74,75,65)(69,73,77,70);(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)(62,65,67,75,68,74,66,64)(63,70,71,69,72,73,76,77);(54,56,57,55)(59,61)(62,64,67,65,68,75,66,74)(63,69,71,73,72,77,76,70);(54,55,57,56)(58,59,60,61)(62,63)(64,73)(65,69)(66,71)(67,76)(68,72)(70,75)(74,77)

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 173416);