#27537289

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

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33436

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	D₄×C₁₆×C₅₃	C₂
prime radical	#16811132	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	#33436
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322765);