#27351608

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17596

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#533	D₅₃
hypocenter	C₅₃	#17596
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#533

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 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)(54,58,59,57)(55,61,62,67)(56,65,66,64)(60,68,63,69)(71,76)(75,84)(77,81)(78,85);(54,57,59,58)(55,61,62,67)(56,64,66,65)(60,68,63,69)(71,76)(75,84)(77,81)(78,85);(54,56,59,66)(55,60,62,63)(57,64,58,65)(61,68,67,69)(70,72,74,80)(71,75,76,84)(73,82,83,79)(77,85,81,78);(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)(56,63,66,60)(57,61)(58,67)(59,62)(64,69,65,68)(70,71)(72,78)(73,81)(74,76)(75,79)(77,83)(80,85)(82,84)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=gh,d^2=gh,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,i^a=i^52,e^b=eh,f^b=fgh

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, 137084);