#27432073

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2051 ⋊ C₂
Sylow	C₅₃ ⋊ #25189

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2051	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	#2051	D₅₃
hypocenter	C₅₃	#25189
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2051

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,61,64,76)(59,67,69,84)(60,71,73,88)(62,79,63,75)(65,80,66,82)(68,83,77,74)(70,85,81,89)(72,87,78,86);(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)(55,57)(58,60)(59,66)(61,71)(62,72)(63,78)(64,73)(65,69)(67,82)(68,81)(70,77)(74,85)(75,86)(76,88)(79,87)(80,84)(83,89);(54,55)(56,57)(58,60,63,72,64,73,62,78)(59,65,68,81,69,66,77,70)(61,71,75,87,76,88,79,86)(67,80,83,89,84,82,74,85);(58,59,63,68,64,69,62,77)(60,70,72,65,73,81,78,66)(61,74,75,67,76,83,79,84)(71,80,87,89,88,82,86,85)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 217549);