#27539254

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3176 ⋊ C₂
Sylow	C₅₃ ⋊ #33682

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3176	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#3176	D₅₃
hypocenter	C₅₃	#33682
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

580 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)(54,56,57,60)(55,59,58,61)(64,74)(68,73)(75,89)(76,90)(81,86)(82,87)(83,88)(91,93);(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)(63,69)(66,79)(67,78)(68,81)(70,84)(71,72)(73,86)(75,91)(76,90)(77,92)(80,85)(82,83)(87,88)(89,93);(55,58)(59,61)(62,64)(63,68)(65,74)(66,75)(67,76)(69,81)(70,73)(71,82)(72,83)(77,89)(78,90)(79,91)(80,88)(84,86)(85,87)(92,93);(54,55)(56,59)(57,58)(60,61)(62,63)(64,73)(65,70)(66,69)(67,80)(68,74)(71,79)(72,78)(75,86)(76,83)(77,84)(81,89)(82,93)(85,92)(87,91)(88,90)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324730);