#27290967

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#101×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #11035

Subgroups

	subgroup	G/H
center	C₂²	#16810539
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#101×C₅₃	C₂
socle	C₂²×C₅₃	#394
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#101	D₅₃
hypocenter	C₅₃	#11035
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#101

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(58,61)(59,66)(60,70)(62,78)(63,77)(64,76)(65,80)(67,74)(68,85)(69,86)(71,88)(72,75)(73,87)(79,83)(81,89)(82,84);(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)(58,60,64,73)(59,65,67,69)(61,75,76,88)(62,72,63,71)(66,82,74,89)(68,84,79,81)(70,78,87,77)(80,85,86,83);(54,55,56,57)(58,59,63,68,64,67,62,79)(60,69,72,84,73,65,71,81)(61,74,78,85,76,66,77,83)(70,80,88,82,87,86,75,89)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76443);