#27539330

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2180 ⋊ C₂
Sylow	C₅₃ ⋊ #33689

Subgroups

	subgroup	G/H
center	C₂	#20284804
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2180	C₂
socle	C₂×C₅₃	#3109
prime radical	#16811420	C₂²
hypercenter	#2180	D₅₃
hypocenter	C₅₃	#33689
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #2180

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324806);