#27276077

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1479 ⋊ C₂
Sylow	C₅₃ ⋊ #8431

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1479	C₂
socle	C₂²×C₅₃	#379
prime radical	#16811173	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1479	D₅₃
hypocenter	C₅₃	#8431
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1479

696 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1187 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61553);