#27433303

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2090 ⋊ C₂
Sylow	C₅₃ ⋊ #25271

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2090	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811135	C₂²
hypercenter	#2090	D₅₃
hypocenter	C₅₃	#25271
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2090

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(55,58)(59,61)(62,66,67,68)(63,71,72,81)(64,74,75,82)(65,78,80,79)(69,84,85,93)(70,87,88,76)(73,90,77,91)(83,92,86,89);(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)(54,56,57,60)(55,59,58,61)(62,65,67,80)(63,70,72,88)(64,73,75,77)(66,78,68,79)(69,83,85,86)(71,87,81,76)(74,90,82,91)(84,92,93,89);(55,58)(59,61)(62,64,66,74,67,75,68,82)(63,69,71,84,72,85,81,93)(65,77,79,90,80,73,78,91)(70,86,76,92,88,83,87,89);(54,55)(56,59)(57,58)(60,61)(62,63,68,81,67,72,66,71)(64,69,82,93,75,85,74,84)(65,76,78,70,80,87,79,88)(73,89,90,83,77,92,91,86)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 218779);