#27537401

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2176 ⋊ C₂
Sylow	C₅₃ ⋊ #33451

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2176	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811056	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2176	D₅₃
hypocenter	C₅₃	#33451
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322877);