#27276523

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1500 ⋊ C₂
Sylow	C₅₃ ⋊ #8504

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1500	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288854	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1500	D₅₃
hypocenter	C₅₃	#8504
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1500

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(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)(56,61)(58,60)(62,65)(63,69)(64,73)(66,80)(67,77)(68,82)(70,84)(71,87)(72,88)(74,92)(75,90)(76,85)(78,81)(79,91)(83,89)(86,93);(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)(54,55)(56,60)(57,59)(58,61)(62,64)(63,68)(65,73)(66,79)(67,75)(69,82)(70,72)(71,86)(74,81)(76,89)(77,90)(78,92)(80,91)(83,85)(84,88)(87,93);(55,58,59,60)(62,63)(64,72,75,68)(65,76,77,87)(66,71)(67,70)(69,78,84,80)(73,89)(74,86,79,83)(81,85)(82,91)(88,92)(90,93)

Polycyclic presentation:

2^8,53|a^2=f,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fg,i^c=i^52

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61999);