#27365130

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18972

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3555	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#121×C₅₃	D₄
hypercenter	#3555	D₅₃
hypocenter	C₅₃	#18972
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3555

818 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)(56,62)(60,67)(63,68)(66,69)(70,74,76,75)(71,79,81,80)(72,83,84,88)(73,86,87,89)(77,91,92,94)(78,85,93,95)(82,97,98,99)(90,100,101,96);(54,56,58,62)(55,60,61,67)(57,63,64,68)(59,66,65,69)(70,73)(71,78)(72,82)(74,86)(75,89)(76,87)(77,90)(79,85)(80,95)(81,93)(83,97)(84,98)(88,99)(91,100)(92,101)(94,96);(54,55,58,61)(56,60,62,67)(57,65,64,59)(63,69,68,66)(70,72)(71,77)(73,82)(74,83)(75,88)(76,84)(78,90)(79,91)(80,94)(81,92)(85,100)(86,97)(87,98)(89,99)(93,101)(95,96);(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)(55,59)(57,64)(60,66)(61,65)(63,68)(67,69)(70,71,76,81)(72,77,84,92)(73,85,87,95)(74,80,75,79)(78,89,93,86)(82,96,98,100)(83,94,88,91)(90,97,101,99)

Polycyclic presentation:

2^8,53|a^2=h,b^2=g,c^2=g,d^2=h,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,d^c=dg,e^c=eh,f^c=fh

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150606);