#27386106

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 ⋊ #21014

Subgroups

	subgroup	G/H
center	C₄	#16810499
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#410	C₂
socle	C₂×C₅₃	C₂×#386
prime radical	#20288044	C₂
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#410	D₅₃
hypocenter	C₅₃	#21014
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#410

1748 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)(54,58,59,70)(55,62,63,75)(56,65,66,78)(57,68,69,81)(60,72,73,83)(61,74,67,80)(64,76,77,84)(71,82,79,85);(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)(58,68)(59,69)(60,71)(62,74)(63,67)(65,76)(66,77)(70,81)(72,82)(73,79)(75,80)(78,84)(83,85);(54,56)(55,60,63,73)(57,64)(58,65)(59,66)(61,71,67,79)(62,72,75,83)(68,76)(69,77)(70,78)(74,82,80,85)(81,84);(54,55)(56,60,66,73)(57,67,69,61)(58,62)(59,63)(64,71)(65,72,78,83)(68,80,81,74)(70,75)(76,82)(77,79)(84,85)

Polycyclic presentation:

2^8,53|a^2=g,b^2=eg,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,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, 171582);