#27292900

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11361

Subgroups

	subgroup	G/H
center	C₄	#16810510
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1740	C₂
socle	C₂×C₅₃	#1445
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1740	D₅₃
hypocenter	C₅₃	#11361
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #1740

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 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)(54,57)(55,62,65,80)(56,67)(58,73)(59,74)(60,75)(61,76,77,83)(63,71,64,79)(66,82,81,78)(68,72)(69,84)(70,85);(54,56)(55,61,65,77)(57,72)(58,68)(59,69)(60,70)(62,78,80,82)(63,66,64,81)(67,74)(71,76,79,83)(73,85)(75,84);(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)(54,55,59,63,60,65,58,64)(56,66,69,77,70,81,68,61)(57,71,73,62,75,79,74,80)(67,83,72,82,85,76,84,78)

Polycyclic presentation:

2^8,53|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^52,c^b=cf,f^b=fh,f^c=fh,g^c=gh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78376);