#27411034

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2003 ⋊ C₂
Sylow	C₅₃ ⋊ #23403

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2003	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#16811419	C₂
hypercenter	#2003	D₅₃
hypocenter	C₅₃	#23403
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2003

1942 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)(58,62,64,63)(59,67,68,76)(60,71,72,77)(61,73,75,74)(65,79,80,69)(66,82,83,84)(70,86,87,88)(78,85,81,89);(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)(58,61,64,75)(59,66,68,83)(60,70,72,87)(62,74,63,73)(65,78,80,81)(67,84,76,82)(69,85,79,89)(71,88,77,86);(58,60)(59,65)(61,70)(62,71)(63,77)(64,72)(66,81)(67,79)(68,80)(69,76)(73,86)(74,88)(75,87)(78,83)(82,89)(84,85);(54,55)(56,57)(58,59)(60,69)(61,66)(62,67)(63,76)(64,68)(65,71)(70,85)(72,79)(73,82)(74,84)(75,83)(77,80)(78,88)(81,86)(87,89)

Polycyclic presentation:

2^8,53|c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gh,c^b=cg,e^b=eh,d^c=dh,e^c=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, 196510);