#27301500

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1385 ⋊ C₂
Sylow	C₅₃ ⋊ #13068

Subgroups

	subgroup	G/H
center	C₂	#20275142
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1385	C₂
socle	C₂×C₅₃	#1893
prime radical	C₂×#16810894	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1385	D₅₃
hypocenter	C₅₃	#13068
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

470 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86976);