#27365898

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3238 ⋊ C₂
Sylow	C₅₃ ⋊ #19051

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3238	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289067	C₂
hypercenter	#3238	D₅₃
hypocenter	C₅₃	#19051
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3238

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(65,76)(69,84)(73,90)(77,80)(82,93)(85,87)(88,89)(91,92);(54,55)(56,61)(57,59)(58,60)(62,65)(63,69)(64,73)(66,80)(67,76)(68,82)(70,87)(71,84)(72,88)(74,92)(75,90)(77,81)(78,85)(79,91)(83,93)(86,89);(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)(56,60)(58,61)(62,64)(63,68)(65,73)(66,79)(67,75)(69,82)(70,86)(71,83)(72,78)(74,81)(76,90)(77,92)(80,91)(84,93)(85,88)(87,89);(55,58)(56,60)(59,61)(62,63,67,71)(64,72,75,86)(65,69,76,84)(66,78,81,70)(68,79,83,74)(73,89,90,88)(77,87,80,85)(82,92,93,91)

Polycyclic presentation:

2^8,53|a^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,i^b=i^52,d^c=dg,e^c=eh,f^c=fgh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 151374);