#27496223

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #30235

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1992	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#1992	D₅₃
hypocenter	C₅₃	#30235
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1992

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 281699);