#27375025

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #19861

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3303	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	#3303	D₅₃
hypocenter	C₅₃	#19861
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3303

596 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 160501);