#27374080

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

Aliases

No aliases for this group.

Splittings

Fitting	Q₈×SD₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #19798

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	Q₈×SD₁₆×C₅₃	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#16810818	C₂²
hypercenter	Q₈×SD₁₆	D₅₃
hypocenter	C₅₃	#19798
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈×SD₁₆

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 159556);