#27541132

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3342 ⋊ C₂
Sylow	C₅₃ ⋊ #33878

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3342	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20286402	C₂
hypercenter	#3342	D₅₃
hypocenter	C₅₃	#33878
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

562 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)(56,61)(58,60)(62,64,68,67)(63,69,72,83)(65,73,78,76)(66,74,82,81)(70,84,86,85)(71,79,88,93)(75,89,91,87)(77,80,90,92);(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)(63,70)(64,73)(66,80)(67,76)(68,78)(69,84)(71,87)(72,86)(74,90)(75,79)(77,81)(82,92)(83,85)(88,89)(91,93);(54,56)(55,58)(57,61)(59,60)(62,64,68,67)(63,69,72,83)(65,76,78,73)(66,74,82,81)(70,85,86,84)(71,79,88,93)(75,87,91,89)(77,92,90,80);(54,55,57,59)(56,60,61,58)(62,63)(64,69)(65,75)(66,79)(67,83)(68,72)(70,77)(71,81)(73,89)(74,88)(76,87)(78,91)(80,84)(82,93)(85,92)(86,90)

Polycyclic presentation:

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

Conjugacy classes & character table

953 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326608);