#27387971

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1969 ⋊ C₂
Sylow	C₅₃ ⋊ #21232

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1969	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1969	D₅₃
hypocenter	C₅₃	#21232
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1969

1470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 173447);