#27415604

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1793 ⋊ C₂
Sylow	C₅₃ ⋊ #23891

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1793	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	#1793	D₅₃
hypocenter	C₅₃	#23891
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1793

1622 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)(58,61,63,77)(59,66,68,82)(60,71,73,87)(62,79,64,76)(65,80,81,89)(67,83,78,75)(69,84,70,85)(72,88,74,86);(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)(55,57)(58,59)(60,70)(61,66)(62,67)(63,68)(64,78)(65,72)(69,73)(71,85)(74,81)(75,76)(77,82)(79,83)(80,88)(84,87)(86,89);(58,60,63,73)(59,65,68,81)(61,71,77,87)(62,74,64,72)(66,80,82,89)(67,70,78,69)(75,84,83,85)(76,88,79,86);(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)(58,59,63,68)(60,69,73,70)(61,75,77,83)(62,78,64,67)(65,74,81,72)(66,79,82,76)(71,80,87,89)(84,88,85,86)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201080);