#27536832

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2120 ⋊ C₂
Sylow	C₅₃ ⋊ #33347

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2120	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#20288045	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2120	D₅₃
hypocenter	C₅₃	#33347
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

1236 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)(54,57,59,69)(55,61,63,75)(56,66,65,78)(58,68,71,82)(60,73,72,83)(62,74,70,81)(64,77,76,84)(67,80,79,85);(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)(56,65)(60,72)(64,76)(66,78)(67,79)(73,83)(77,84)(80,85);(54,56)(55,60)(57,66)(58,67)(59,65)(61,73)(62,64)(63,72)(68,80)(69,78)(70,76)(71,79)(74,77)(75,83)(81,84)(82,85);(54,55)(56,64)(57,61)(58,70,71,62)(59,63)(60,67,72,79)(65,76)(66,77)(68,81,82,74)(69,75)(73,80,83,85)(78,84)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322308);