#27352245

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1534 ⋊ C₂
Sylow	C₅₃ ⋊ #17680

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1534	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#20288687	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1534	D₅₃
hypocenter	C₅₃	#17680
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1534

878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137721);