#27537740

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

Aliases

No aliases for this group.

Splittings

Fitting	Q₈×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33498

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	Q₈×C₁₆×C₅₃	C₂
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	#33498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323216);