#27435523

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #25438

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3057	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₅₃×#535	C₂²
hypercenter	#3057	D₅₃
hypocenter	C₅₃	#25438
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3057

892 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:

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=ef,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,i^b=i^52,d^c=dh,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, 220999);