#27469945

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #27991

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2086	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2086	D₅₃
hypocenter	C₅₃	#27991
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2086

590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 255421);