#27299726

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2133 ⋊ C₂
Sylow	C₅₃ ⋊ #12729

Subgroups

	subgroup	G/H
center	C₂	#20274631
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×Q₈	C₂²×D₅₃
Fitting	C₅₃×#2133	C₂
socle	C₂×C₅₃	#1782
prime radical	#16811187	C₄
cosocle	C₂²×Q₈×C₅₃	C₂³
hypercenter	#2133	D₅₃
hypocenter	C₅₃	#12729
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×Q₈ ◁ #2133

690 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 103 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)(54,56)(55,61,64,65)(57,66)(58,68)(59,67)(60,73,62,76)(63,77,79,80)(69,81)(70,82)(71,75)(72,83)(74,84,78,85);(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)(55,60)(57,69)(58,71)(61,73)(62,64)(63,74)(65,76)(66,81)(68,75)(77,84)(78,79)(80,85);(54,55,58,63,59,64,71,79)(56,65,68,80,67,61,75,77)(57,60,70,74,69,62,72,78)(66,76,82,85,81,73,83,84)

Polycyclic presentation:

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

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85202);