#27273715

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7916

Subgroups

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7916
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

622 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)(58,61,62,69)(59,64,65,71)(60,67,68,73)(63,70,66,72)(76,83)(79,82)(84,85)(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)(58,60)(59,63)(61,67)(62,68)(64,70)(65,66)(69,73)(71,72)(74,76)(75,79)(77,85)(78,83)(80,82)(81,89)(84,86)(87,88);(58,59)(60,66,68,63)(61,64)(62,65)(67,72,73,70)(69,71)(74,75,77,81,78,80,86,88)(76,82,84,89,83,79,85,87)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=gh,e^2=h,b^a=bd,d^a=dg,c^b=ch,e^b=eg,i^b=i^52

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59191);