#27364900

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3249 ⋊ C₂
Sylow	C₅₃ ⋊ #18953

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3249	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811173	C₂²
hypercenter	#3249	D₅₃
hypocenter	C₅₃	#18953
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3249

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 101 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)(71,78)(76,91)(77,93)(79,86)(83,95)(90,101)(92,94)(96,100);(54,57,60,59)(55,62,64,68)(56,66,58,67)(61,63,69,65)(70,73)(71,77)(72,80)(74,88)(75,84)(76,90)(78,93)(79,95)(81,99)(82,97)(83,86)(85,89)(87,98)(91,101)(92,100)(94,96);(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)(70,72,75,82)(71,76,78,91)(73,80,84,97)(74,87,89,81)(77,90,93,101)(79,94,86,92)(83,100,95,96)(85,99,88,98);(54,55,60,64)(56,65,58,63)(57,62,59,68)(61,67,69,66)(70,71)(72,76)(73,83)(74,86)(75,78)(77,85)(79,89)(80,96)(81,94)(82,91)(84,95)(87,92)(88,93)(90,98)(97,100)(99,101)

Polycyclic presentation:

2^8,53|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,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, 150376);