#27403047

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #22741

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1949	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1949	D₅₃
hypocenter	C₅₃	#22741
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1949

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

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

Conjugacy classes & character table

1058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 188523);