#27360260

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1491 ⋊ C₂
Sylow	C₅₃ ⋊ #18536

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1491	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20284807	C₂
hypercenter	#1491	D₅₃
hypocenter	C₅₃	#18536
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1491

940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(55,62)(60,69)(61,66)(63,68)(71,78)(76,79)(77,85)(80,84);(54,57,59,58)(55,61,62,66)(56,64,65,67)(60,68,69,63)(70,72,75,83)(71,77,78,85)(73,82,74,81)(76,84,79,80);(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)(54,56,59,65)(55,60,62,69)(57,64,58,67)(61,68,66,63)(71,76)(73,74)(77,84)(78,79)(80,85)(81,82);(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)(56,63)(57,61)(58,66)(59,62)(60,64)(65,68)(67,69)(70,71)(72,80)(73,76)(74,79)(75,78)(77,82)(81,85)(83,84)

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 145736);