#27292684

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11320

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133203
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#341	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#341	D₅₃
hypocenter	C₅₃	#11320
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

546 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)(58,61,62,75,63,76,69,84)(59,65,66,81,67,73,78,87)(60,70,71,85,72,74,77,89)(64,79,68,83,80,86,82,88);(58,60,69,71,63,72,62,77)(59,64,78,68,67,80,66,82)(61,74,84,89,76,70,75,85)(65,79,87,83,73,86,81,88);(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)(58,59)(60,68,72,82)(61,73)(62,66)(63,67)(64,71,80,77)(65,76)(69,78)(70,83,74,88)(75,87)(79,85,86,89)(81,84)

Polycyclic presentation:

2^8,53|a^2=g,b^2=de,c^2=d,d^2=h,b^a=bd,c^a=ce,f^a=fh,i^a=i^52,c^b=cf,d^b=dh,e^b=eh,g^b=gh,f^c=fh,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, 78160);