#27291941

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#341 ⋊ C₂
Sylow	C₅₃ ⋊ #11194

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#341	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#341	D₅₃
hypocenter	C₅₃	#11194
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 77417);