#27299247

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12623

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2120	C₂
socle	C₂×C₅₃	C₄×#99
prime radical	C₅₃×#525	C₂²
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#2120	D₅₃
hypocenter	C₅₃	#12623
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

470 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)(54,57,58,69,60,55,62,63)(56,65,67,78,66,61,73,74)(59,70,71,82,72,64,76,77)(68,79,81,85,80,75,83,84);(54,56,59,68)(55,61,64,75)(57,65,70,79)(58,67,71,81)(60,66,72,80)(62,73,76,83)(63,74,77,84)(69,78,82,85);(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)(54,55,58,63,60,57,62,69)(56,65,67,78,66,61,73,74)(59,64,71,77,72,70,76,82)(68,75,81,84,80,79,83,85)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=e,e^2=h,b^a=bd,f^a=fg,i^a=i^52,d^b=dg,g^b=gh,i^c=i^52,f^d=fh

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84723);