#27363484

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3235 ⋊ C₂
Sylow	C₅₃ ⋊ #18827

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3235	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288495	C₂
hypercenter	#3235	D₅₃
hypocenter	C₅₃	#18827
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3235

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 148960);