#27350108

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17385

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17385
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

988 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)(55,62,63,61)(56,66,67,65)(60,69,64,68)(70,73,75,74)(71,77,79,78)(72,81,80,82)(76,84,83,85);(54,57,59,58)(55,61,63,62)(56,65,67,66)(60,68,64,69)(72,80)(76,83)(81,82)(84,85);(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)(54,56,59,67)(55,60,63,64)(57,65,58,66)(61,68,62,69)(70,72)(71,76)(73,81)(74,82)(75,80)(77,84)(78,85)(79,83);(54,55)(56,64,67,60)(57,61)(58,62)(59,63)(65,69,66,68)(70,71,74,78,75,79,73,77)(72,76,82,85,80,83,81,84)

Polycyclic presentation:

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

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, 135584);