#27537283

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2168 ⋊ C₂
Sylow	C₅₃ ⋊ #33435

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2168	C₂
prime radical	#16811170	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2168	D₅₃
hypocenter	C₅₃	#33435
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2168

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 points:

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=f,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=dh,d^b=dh,i^d=i^52

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322759);