#27292295

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11259

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1735	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1735	D₅₃
hypocenter	C₅₃	#11259
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1735

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(55,57)(58,61)(59,65,66,78)(60,69)(62,73)(63,74)(64,76,68,80)(67,72,79,87)(70,83)(71,84)(75,88)(77,86,81,82)(85,89);(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)(56,57)(58,60,62,70,63,71,75,85)(59,64,67,77,66,68,79,81)(61,69,73,83,74,84,88,89)(65,76,72,86,78,80,87,82);(58,59)(60,68,71,64)(61,72,74,87)(62,67)(63,66)(65,73,78,88)(69,82)(70,81,85,77)(75,79)(76,83)(80,89)(84,86)

Polycyclic presentation:

2^8,53|a^2=g,b^2=e,c^2=d,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,i^b=i^52,g^c=gh,i^c=i^52

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 77771);