#27365138

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1648 ⋊ C₂
Sylow	C₅₃ ⋊ #18973

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1648	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#13133192	D₄
hypercenter	#1648	D₅₃
hypocenter	C₅₃	#18973
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1648

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150614);