#27273483

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7851

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂×C₄²
Fitting	C₂×C₅₃×#431	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7851
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 89 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)(59,64)(60,66)(65,71)(67,72)(75,80)(79,85)(81,87)(86,89);(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)(55,57)(58,60)(59,63)(61,67)(62,66)(64,70)(65,69)(68,72)(71,73)(74,76,77,82,78,83,84,88)(75,79,81,86,80,85,87,89);(54,55)(56,57)(58,59,61,65,62,64,68,71)(60,63,67,69,66,70,72,73)(74,75)(76,79)(77,81)(78,80)(82,86)(83,85)(84,87)(88,89)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,e^2=g,f^2=h,b^a=bd,c^a=cgh,c^b=cg,i^b=i^52

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 58959);