#27350019

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17370

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#520	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₅₃	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#520	D₅₃
hypocenter	C₅₃	#17370
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×#520

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 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)(62,65,66,73)(63,68,69,75)(64,71,72,77)(67,74,70,76);(55,58)(59,61)(65,73)(68,75)(71,77)(74,76);(62,64,66,72)(63,67,69,70)(65,71,73,77)(68,74,75,76);(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)(54,55,56,59,57,58,60,61)(62,63)(64,70,72,67)(65,68)(66,69)(71,76,77,74)(73,75)

Polycyclic presentation:

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

Conjugacy classes & character table

1826 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135495);