#27472343

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #28170

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2074	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2074	D₅₃
hypocenter	C₅₃	#28170
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2074

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 97 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)(58,60,61,65)(59,63,62,64)(68,77)(72,86)(76,91)(78,93)(85,96)(87,94)(90,97)(92,95);(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)(55,57)(66,69,71,81)(67,73,75,88)(68,76,77,91)(70,83,84,80)(72,85,86,96)(74,89,82,79)(78,95,93,92)(87,97,94,90);(54,55,56,57)(59,62)(63,64)(66,68)(67,72)(69,76)(70,78)(71,77)(73,85)(74,87)(75,86)(79,90)(80,92)(81,91)(82,94)(83,95)(84,93)(88,96)(89,97);(58,59)(60,64)(61,62)(63,65)(66,67,71,75)(68,72,77,86)(69,79,81,89)(70,82,84,74)(73,83,88,80)(76,90,91,97)(78,94,93,87)(85,95,96,92)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 257819);