#27590778

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#125×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #36698

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₄×#125×C₅₃	C₂
prime radical	C₂²×D₂₁₂	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#36698
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 376254);