#27537429

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×SD₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33454

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×SD₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811060	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×SD₃₂	D₅₃
hypocenter	C₅₃	#33454
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=f,e^2=gh,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322905);