#27547180

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #34387

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₂×C₅₃×#529	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂×#121×C₅₃	C₂²
hypercenter	C₂×#529	D₅₃
hypocenter	C₅₃	#34387
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#529

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(59,65)(63,66)(64,71)(70,72)(74,77,78,84)(75,81,80,87)(76,83,82,88)(79,86,85,89);(60,68)(63,66)(67,73)(70,72)(75,80)(79,85)(81,87)(86,89);(58,61,62,69)(59,64,65,71)(60,67,68,73)(63,70,66,72)(75,80)(76,82)(81,87)(83,88);(58,60)(59,63)(61,67)(62,68)(64,70)(65,66)(69,73)(71,72)(74,76)(75,79)(77,83)(78,82)(80,85)(81,86)(84,88)(87,89);(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)(54,55,56,57)(58,59)(60,66)(61,64)(62,65)(63,68)(67,72)(69,71)(70,73)(74,75)(76,79)(77,81)(78,80)(82,85)(83,86)(84,87)(88,89)

Polycyclic presentation:

2^8,53|a^2=h,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,i^a=i^52,c^b=cg,d^b=df

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 332656);