#27415715

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#536 ⋊ C₂
Sylow	C₅₃ ⋊ #23905

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#536	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288359	C₂
hypercenter	C₂×#536	D₅₃
hypocenter	C₅₃	#23905
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#536

1744 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)(59,66)(60,69)(61,71)(63,74)(64,76)(65,78)(67,73)(68,81)(70,84)(72,85)(75,87)(77,83)(79,86)(80,88)(82,89);(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)(55,57)(58,61)(59,65)(60,68)(62,71)(63,72)(64,75)(66,78)(67,79)(69,81)(70,82)(73,86)(74,85)(76,87)(77,80)(83,88)(84,89);(54,55)(56,57)(58,60,63,70)(59,64,67,77)(61,68,72,82)(62,69,74,84)(65,75,79,80)(66,76,73,83)(71,81,85,89)(78,87,86,88);(58,59,63,67)(60,64,70,77)(61,65,72,79)(62,73,74,66)(68,80,82,75)(69,83,84,76)(71,78,85,86)(81,88,89,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201191);