#27354950

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1587 ⋊ C₂
Sylow	C₅₃ ⋊ #17991

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1587	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1587	D₅₃
hypocenter	C₅₃	#17991
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1587

544 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)(55,60)(56,62)(58,66)(59,65)(61,64)(63,69)(67,68)(71,78)(73,84)(76,90)(79,86)(80,93)(81,95)(85,97)(96,99);(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)(54,56,58,61,59,63,67,55)(57,62,66,64,65,69,68,60)(70,73,75,84)(71,77,78,92)(72,81,83,95)(74,85,88,97)(76,89,90,100)(79,91,86,98)(80,94,93,101)(82,96,87,99);(57,65)(60,64)(62,69)(66,68)(70,72,75,83)(71,76,78,90)(73,81,84,95)(74,87,88,82)(77,89,92,100)(79,93,86,80)(85,99,97,96)(91,101,98,94);(54,55,58,56,59,61,67,63)(57,64,66,69,65,60,68,62)(70,71)(72,80)(73,77)(74,86)(75,78)(76,82)(79,88)(81,94)(83,93)(84,92)(85,98)(87,90)(89,96)(91,97)(95,101)(99,100)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dgh,e^a=eg,d^b=dh,e^b=eg,d^c=dg,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 140426);