#27350281

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#323 ⋊ C₂
Sylow	C₅₃ ⋊ #17412

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#323	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#13133186	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#323	D₅₃
hypocenter	C₅₃	#17412
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#323

1046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

1199 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135757);