#27538393

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2210 ⋊ C₂
Sylow	C₅₃ ⋊ #33584

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2210	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811057	C₂
hypercenter	#2210	D₅₃
hypocenter	C₅₃	#33584
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2210

722 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:

(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)(58,61,62,64)(59,67,65,69)(60,71,72,79)(63,74,75,78)(66,82,80,85)(68,83,81,76)(70,84,86,87)(73,77,88,89);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,60)(59,66)(61,71)(62,72)(63,77)(64,79)(65,80)(67,82)(68,84)(69,85)(70,76)(73,78)(74,88)(75,89)(81,87)(83,86);(54,55,56,57)(59,65)(66,80)(67,69)(68,81)(70,86)(76,83)(82,85)(84,87);(55,57)(58,59)(60,70)(61,69)(62,65)(63,76)(64,67)(66,73)(68,78)(71,87)(72,86)(74,81)(75,83)(77,85)(79,84)(80,88)(82,89)

Polycyclic presentation:

2^8,53|b^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh,i^d=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323869);