#27386004

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#410 ⋊ C₂
Sylow	C₅₃ ⋊ #21000

Subgroups

	subgroup	G/H
center	C₄	C₂×#13133190
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#410	C₂
socle	C₂×C₅₃	C₂×#386
prime radical	C₂×#16811421	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#410	D₅₃
hypocenter	C₅₃	#21000
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#410

1704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171480);