#27365201

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1495 ⋊ C₂
Sylow	C₅₃ ⋊ #18977

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1495	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289106	C₂
hypercenter	#1495	D₅₃
hypocenter	C₅₃	#18977
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1495

866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150677);