#27299830

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12750

Subgroups

	subgroup	G/H
center	C₂	#20274646
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#576	C₂
socle	C₂×C₅₃	#1786
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#576	D₅₃
hypocenter	C₅₃	#12750
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#576

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 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)(54,56)(55,62)(57,67)(58,75)(59,68)(60,69)(61,77)(63,79)(64,82)(65,81)(66,76)(70,78)(71,72)(73,84)(74,85)(80,83);(54,56)(55,61)(57,71)(58,68)(59,75)(60,69)(62,77)(63,81)(64,78)(65,79)(66,83)(67,72)(70,82)(73,85)(74,84)(76,80);(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)(56,66,69,82)(57,70)(58,64)(59,76)(60,65)(61,67,79,85)(62,68,81,75)(63,73)(71,78,84,80)(72,77)(74,83)

Polycyclic presentation:

2^8,53|a^2=f,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,d^b=dgh,f^b=fh,g^b=gh,e^c=eh,f^c=fh

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85306);