#27385902

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #20986

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#410	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#410	D₅₃
hypocenter	C₅₃	#20986
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#410

1718 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:

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=eg,c^2=h,d^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dh,i^c=i^52

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171378);