#27279559

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₈×#41×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #9095

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#13133242	C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#9095
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 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)(62,64)(63,68)(65,73)(66,71)(67,75)(69,76)(70,77)(72,74);(54,56,57,61,58,55,59,60)(62,64)(63,67,69,77)(65,73)(66,71)(68,75,76,70)(72,74);(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)(54,55,57,60,58,56,59,61)(62,63)(64,67,71,77)(65,70)(66,69)(68,72,76,73)(74,75)

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 65035);