#27536730

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2108 ⋊ C₂
Sylow	C₅₃ ⋊ #33324

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33324
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2108

826 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)(59,65)(64,74)(66,77)(67,78)(75,84)(76,85)(79,87)(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)(60,68)(64,74)(69,80)(70,81)(75,84)(76,85)(82,88)(86,89);(54,55,56,57)(58,60)(59,64)(61,69)(62,70)(63,68)(65,74)(66,75)(67,76)(71,82)(72,80)(73,81)(77,84)(78,85)(79,86)(83,88)(87,89);(58,59,61,66,62,67,71,79,63,65,72,77,73,78,83,87)(60,64,69,75,70,76,82,86,68,74,80,84,81,85,88,89)

Polycyclic presentation:

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

Conjugacy classes & character table

2216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322206);