#27385736

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1740 ⋊ C₂
Sylow	C₅₃ ⋊ #20962

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1740	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂²×#11719945	C₂²
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1740	D₅₃
hypocenter	C₅₃	#20962
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1740

1742 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)(58,62,63,75)(59,66,67,74)(60,69,70,83)(61,72,73,86)(64,77,78,82)(65,79,71,85)(68,80,81,89)(76,87,84,88);(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)(59,65)(60,68)(62,72)(63,73)(64,76)(66,79)(67,71)(69,80)(70,81)(74,85)(75,86)(77,87)(78,84)(82,88)(83,89);(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)(54,56,57,55)(58,60,63,70)(59,64,67,78)(61,68,73,81)(62,69,75,83)(65,76,71,84)(66,77,74,82)(72,80,86,89)(79,87,85,88);(54,55,57,56)(58,59)(60,64)(61,71,73,65)(62,74)(63,67)(66,75)(68,76,81,84)(69,82)(70,78)(72,79,86,85)(77,83)(80,88,89,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171212);