#27537726

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33495

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	D₄×C₁₆×C₅₃	C₂
prime radical	#16811170	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	#33495
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,f^2=g,g^2=h,b^a=be,d^a=de,c^b=ceh,d^b=dh,d^c=dh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323202);