#27351617

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17597

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3519	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3519	D₅₃
hypocenter	C₅₃	#17597
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #3519

1060 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)(57,65)(60,67)(63,69)(66,68);(54,57)(55,60)(56,63)(58,65)(59,66)(61,67)(62,68)(64,69)(71,76)(75,83)(77,84)(78,85);(54,56)(55,59)(57,63)(58,64)(60,66)(61,62)(65,69)(67,68)(70,72)(71,75)(73,81)(74,80)(76,83)(77,85)(78,84)(79,82);(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)(54,55)(56,62,64,59)(57,60)(58,61)(63,68,69,66)(65,67)(70,71)(72,78,80,85)(73,77)(74,76)(75,79,83,81)(82,84)

Polycyclic presentation:

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

Conjugacy classes & character table

1193 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137093);